Volume 12, 2000

University of Bialystok

Copyright (c) 2000 Association of Mizar Users

### The abstract of the Mizar article:

### Ring Ideals

**by****Jonathan Backer,****Piotr Rudnicki, and****Christoph Schwarzweller**- Received November 20, 2000
- MML identifier: IDEAL_1

- [ Mizar article, MML identifier index ]

environ vocabulary RLVECT_1, ALGSTR_1, VECTSP_1, VECTSP_2, BINOM, BINOP_1, LATTICES, REALSET1, FINSEQ_1, FILTER_2, ALGSTR_2, COLLSP, BOOLE, ARYTM_1, GROUP_1, RELAT_1, FUNCT_1, FUNCT_7, ARYTM_3, FINSEQ_4, PRELAMB, MCART_1, SETFAM_1, RLVECT_3, SUBSET_1, GCD_1, LATTICE3, SQUARE_1, NEWTON, FINSET_1, INT_3, WAYBEL_0, TARSKI, NORMSP_1, GR_CY_1, IDEAL_1; notation TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, CQC_SIM1, FINSET_1, FUNCT_1, FINSEQ_1, VECTSP_1, INT_3, NORMSP_1, NUMBERS, XCMPLX_0, XREAL_0, NAT_1, RELSET_1, RLVECT_1, PARTFUN1, FUNCT_2, ALGSTR_1, PRE_TOPC, GR_CY_1, PRE_CIRC, MCART_1, DOMAIN_1, FINSEQ_4, POLYNOM1, SETFAM_1, STRUCT_0, GROUP_1, BINOP_1, VECTSP_2, FUNCT_7, REALSET1, GCD_1, BINOM; constructors INT_3, CQC_SIM1, PRE_CIRC, GROUP_2, ALGSEQ_1, BINOM, GCD_1, ALGSTR_2, DOMAIN_1, POLYNOM1, MONOID_0, BINOP_1, PRE_TOPC; clusters INT_3, RELSET_1, FINSET_1, SUBSET_1, VECTSP_2, STRUCT_0, PRE_TOPC, FINSEQ_1, FINSEQ_5, XBOOLE_0, POLYNOM1, INT_1, GCD_1, BINOM, REALSET1, VECTSP_1, NAT_1, XREAL_0, MEMBERED, RELAT_1, FUNCT_2, PRE_CIRC, NUMBERS, ORDINAL2; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; begin :: Preliminaries definition cluster add-associative left_zeroed right_zeroed (non empty LoopStr); end; definition cluster Abelian left_zeroed right_zeroed add-cancelable well-unital add-associative associative commutative distributive non trivial (non empty doubleLoopStr); end; theorem :: IDEAL_1:1 for V being add-associative left_zeroed right_zeroed (non empty LoopStr), v,u being Element of V holds Sum <* v,u *> = v + u; begin :: Ideals definition let L be non empty LoopStr, F being Subset of L; attr F is add-closed means :: IDEAL_1:def 1 for x, y being Element of L st x in F & y in F holds x+y in F; end; definition let L be non empty HGrStr, F be Subset of L; attr F is left-ideal means :: IDEAL_1:def 2 for p, x being Element of L st x in F holds p*x in F; attr F is right-ideal means :: IDEAL_1:def 3 for p, x being Element of L st x in F holds x*p in F; end; definition let L be non empty LoopStr; cluster add-closed (non empty Subset of L); end; definition let L be non empty HGrStr; cluster left-ideal (non empty Subset of L); cluster right-ideal (non empty Subset of L); end; definition let L be non empty doubleLoopStr; cluster add-closed left-ideal right-ideal (non empty Subset of L); cluster add-closed right-ideal (non empty Subset of L); cluster add-closed left-ideal (non empty Subset of L); end; definition let R be commutative (non empty HGrStr); cluster left-ideal -> right-ideal (non empty Subset of R); cluster right-ideal -> left-ideal (non empty Subset of R); end; definition let L be non empty doubleLoopStr; mode Ideal of L is add-closed left-ideal right-ideal (non empty Subset of L); end; definition let L be non empty doubleLoopStr; mode RightIdeal of L is add-closed right-ideal (non empty Subset of L); end; definition let L be non empty doubleLoopStr; mode LeftIdeal of L is add-closed left-ideal (non empty Subset of L); end; theorem :: IDEAL_1:2 for R being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I being left-ideal (non empty Subset of R) holds 0.R in I; theorem :: IDEAL_1:3 for R being left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), I being right-ideal (non empty Subset of R) holds 0.R in I; theorem :: IDEAL_1:4 for L being right_zeroed (non empty doubleLoopStr) holds {0.L} is add-closed; theorem :: IDEAL_1:5 for L being left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr) holds {0.L} is left-ideal; theorem :: IDEAL_1:6 for L being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr) holds {0.L} is right-ideal; theorem :: IDEAL_1:7 for L being add-associative right_zeroed right_complementable distributive (non empty doubleLoopStr) holds {0.L} is Ideal of L; theorem :: IDEAL_1:8 for L being add-associative right_zeroed right_complementable right-distributive (non empty doubleLoopStr) holds {0.L} is LeftIdeal of L; theorem :: IDEAL_1:9 for L being add-associative right_zeroed right_complementable left-distributive (non empty doubleLoopStr) holds {0.L} is RightIdeal of L; theorem :: IDEAL_1:10 for L being non empty doubleLoopStr holds the carrier of L is Ideal of L; theorem :: IDEAL_1:11 for L being non empty doubleLoopStr holds the carrier of L is LeftIdeal of L; theorem :: IDEAL_1:12 for L being non empty doubleLoopStr holds the carrier of L is RightIdeal of L; definition let R be left_zeroed right_zeroed add-cancelable distributive (non empty doubleLoopStr), I be Ideal of R; redefine attr I is trivial means :: IDEAL_1:def 4 I = {0.R}; end; definition let S be 1-sorted, T be Subset of S; attr T is proper means :: IDEAL_1:def 5 T <> the carrier of S; end; definition let S be non empty 1-sorted; cluster proper Subset of S; end; definition let R be non trivial left_zeroed right_zeroed add-cancelable distributive (non empty doubleLoopStr); cluster proper Ideal of R; end; theorem :: IDEAL_1:13 for L being add-associative right_zeroed right_complementable left-distributive left_unital (non empty doubleLoopStr), I being left-ideal (non empty Subset of L), x being Element of L st x in I holds -x in I; theorem :: IDEAL_1:14 for L being add-associative right_zeroed right_complementable right-distributive right_unital (non empty doubleLoopStr), I being right-ideal (non empty Subset of L), x being Element of L st x in I holds -x in I; theorem :: IDEAL_1:15 for L being add-associative right_zeroed right_complementable left-distributive left_unital (non empty doubleLoopStr), I being LeftIdeal of L, x,y being Element of L st x in I & y in I holds x-y in I; theorem :: IDEAL_1:16 for L being add-associative right_zeroed right_complementable right-distributive right_unital (non empty doubleLoopStr), I being RightIdeal of L, x,y being Element of L st x in I & y in I holds x-y in I; theorem :: IDEAL_1:17 for R being left_zeroed right_zeroed add-cancelable add-associative distributive (non empty doubleLoopStr), I being add-closed right-ideal (non empty Subset of R), a being Element of I, n being Nat holds n*a in I; theorem :: IDEAL_1:18 for R being unital left_zeroed right_zeroed add-cancelable associative distributive (non empty doubleLoopStr), I being right-ideal (non empty Subset of R), a being Element of I, n being Nat st n <> 0 holds a|^n in I; definition let R be non empty LoopStr, I be add-closed (non empty Subset of R); func add|(I,R) -> BinOp of I equals :: IDEAL_1:def 6 (the add of R)|[:I,I:]; end; definition let R be non empty HGrStr, I be right-ideal (non empty Subset of R); func mult|(I,R) -> BinOp of I equals :: IDEAL_1:def 7 (the mult of R)|[:I,I:]; end; definition let R be non empty LoopStr, I be add-closed (non empty Subset of R); func Gr(I,R) -> non empty LoopStr equals :: IDEAL_1:def 8 LoopStr (#I,add|(I,R),In (0.R,I)#); end; definition let R be left_zeroed right_zeroed add-cancelable add-associative distributive (non empty doubleLoopStr), I be add-closed (non empty Subset of R); cluster Gr(I,R) -> add-associative; end; definition let R be left_zeroed right_zeroed add-cancelable add-associative distributive (non empty doubleLoopStr), I be add-closed right-ideal (non empty Subset of R); cluster Gr(I,R) -> right_zeroed; end; definition let R be Abelian (non empty doubleLoopStr), I be add-closed (non empty Subset of R); cluster Gr(I,R) -> Abelian; end; definition let R be Abelian right_unital left_zeroed right_zeroed right_complementable add-associative distributive (non empty doubleLoopStr), I be add-closed right-ideal (non empty Subset of R); cluster Gr(I,R) -> right_complementable; end; theorem :: IDEAL_1:19 for R being right_unital (non empty doubleLoopStr), I being left-ideal (non empty Subset of R) holds I is proper iff not(1_ R in I); theorem :: IDEAL_1:20 for R being left_unital right_unital (non empty doubleLoopStr), I being right-ideal (non empty Subset of R) holds I is proper iff for u being Element of R st u is unital holds not(u in I); theorem :: IDEAL_1:21 for R being right_unital (non empty doubleLoopStr), I being left-ideal right-ideal (non empty Subset of R) holds I is proper iff for u being Element of R st u is unital holds not(u in I); theorem :: IDEAL_1:22 for R being non degenerated comRing holds R is Field iff for I being Ideal of R holds (I = {0.R} or I = the carrier of R); begin :: Linear combinations definition let R be non empty multLoopStr, A be non empty Subset of R; mode LinearCombination of A -> FinSequence of the carrier of R means :: IDEAL_1:def 9 for i being set st i in dom it ex u,v being Element of R, a being Element of A st it/.i = u*a*v; mode LeftLinearCombination of A -> FinSequence of the carrier of R means :: IDEAL_1:def 10 for i being set st i in dom it ex u being Element of R, a being Element of A st it/.i = u*a; mode RightLinearCombination of A -> FinSequence of the carrier of R means :: IDEAL_1:def 11 for i being set st i in dom it ex u being Element of R, a being Element of A st it/.i = a*u; end; definition let R be non empty multLoopStr, A be non empty Subset of R; cluster non empty LinearCombination of A; cluster non empty LeftLinearCombination of A; cluster non empty RightLinearCombination of A; end; definition let R be non empty multLoopStr, A,B be non empty Subset of R, F be LinearCombination of A, G be LinearCombination of B; redefine func F^G -> LinearCombination of (A \/ B); end; theorem :: IDEAL_1:23 for R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be LinearCombination of A holds a*F is LinearCombination of A; theorem :: IDEAL_1:24 for R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be LinearCombination of A holds F*a is LinearCombination of A; theorem :: IDEAL_1:25 for R being right_unital (non empty multLoopStr), A being non empty Subset of R, f being LeftLinearCombination of A holds f is LinearCombination of A; definition let R be non empty multLoopStr, A,B be non empty Subset of R, F be LeftLinearCombination of A, G be LeftLinearCombination of B; redefine func F^G -> LeftLinearCombination of (A \/ B); end; theorem :: IDEAL_1:26 for R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be LeftLinearCombination of A holds a*F is LeftLinearCombination of A; theorem :: IDEAL_1:27 for R be non empty multLoopStr, A be non empty Subset of R, a be Element of R, F be LeftLinearCombination of A holds F*a is LinearCombination of A; theorem :: IDEAL_1:28 for R being left_unital (non empty multLoopStr), A being non empty Subset of R, f being RightLinearCombination of A holds f is LinearCombination of A; definition let R be non empty multLoopStr, A,B be non empty Subset of R, F be RightLinearCombination of A, G be RightLinearCombination of B; redefine func F^G -> RightLinearCombination of (A \/ B); end; theorem :: IDEAL_1:29 for R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be RightLinearCombination of A holds F*a is RightLinearCombination of A; theorem :: IDEAL_1:30 for R be associative (non empty multLoopStr), A be non empty Subset of R, a be Element of R, F be RightLinearCombination of A holds a*F is LinearCombination of A; theorem :: IDEAL_1:31 for R being commutative associative (non empty multLoopStr), A being non empty Subset of R, f being LinearCombination of A holds f is LeftLinearCombination of A & f is RightLinearCombination of A; theorem :: IDEAL_1:32 for S being non empty doubleLoopStr, F being non empty Subset of S, lc being non empty LinearCombination of F ex p being LinearCombination of F, e being Element of S st lc = p^<* e *> & <*e*> is LinearCombination of F; theorem :: IDEAL_1:33 for S being non empty doubleLoopStr, F being non empty Subset of S, lc being non empty LeftLinearCombination of F ex p being LeftLinearCombination of F, e being Element of S st lc = p^<* e *> & <*e*> is LeftLinearCombination of F; theorem :: IDEAL_1:34 for S being non empty doubleLoopStr, F being non empty Subset of S, lc being non empty RightLinearCombination of F ex p being RightLinearCombination of F, e being Element of S st lc = p^<* e *> & <*e*> is RightLinearCombination of F; definition let R be non empty multLoopStr, A be non empty Subset of R, L be LinearCombination of A, E be FinSequence of [:the carrier of R, the carrier of R, the carrier of R:]; pred E represents L means :: IDEAL_1:def 12 len E = len L & for i being set st i in dom L holds L.i = ((E/.i)`1)*((E/.i)`2)*((E/.i)`3) & ((E/.i)`2) in A; end; theorem :: IDEAL_1:35 for R being non empty multLoopStr, A being non empty Subset of R, L be LinearCombination of A ex E be FinSequence of [:the carrier of R, the carrier of R, the carrier of R:] st E represents L; theorem :: IDEAL_1:36 for R, S being non empty multLoopStr, F being non empty Subset of R, lc being LinearCombination of F, G being non empty Subset of S, P being Function of the carrier of R, the carrier of S, E being FinSequence of [:the carrier of R, the carrier of R, the carrier of R:] st P.:F c= G & E represents lc holds ex LC being LinearCombination of G st len lc = len LC & for i being set st i in dom LC holds LC.i = (P.(E/.i)`1)*(P.(E/.i)`2)*(P.(E/.i)`3); definition let R be non empty multLoopStr, A be non empty Subset of R, L be LeftLinearCombination of A, E be FinSequence of [:the carrier of R, the carrier of R:]; pred E represents L means :: IDEAL_1:def 13 len E = len L & for i being set st i in dom L holds L.i = ((E/.i)`1)*((E/.i)`2) & ((E/.i)`2) in A; end; theorem :: IDEAL_1:37 for R being non empty multLoopStr, A being non empty Subset of R, L be LeftLinearCombination of A ex E be FinSequence of [:the carrier of R, the carrier of R:] st E represents L; theorem :: IDEAL_1:38 for R, S being non empty multLoopStr, F being non empty Subset of R, lc being LeftLinearCombination of F, G being non empty Subset of S, P being Function of the carrier of R, the carrier of S, E being FinSequence of [:the carrier of R, the carrier of R:] st P.:F c= G & E represents lc holds ex LC being LeftLinearCombination of G st len lc = len LC & for i being set st i in dom LC holds LC.i = (P.(E/.i)`1)*(P.(E/.i)`2); definition let R be non empty multLoopStr, A be non empty Subset of R, L be RightLinearCombination of A, E be FinSequence of [:the carrier of R, the carrier of R:]; pred E represents L means :: IDEAL_1:def 14 len E = len L & for i being set st i in dom L holds L.i = ((E/.i)`1)*((E/.i)`2) & ((E/.i)`1) in A; end; theorem :: IDEAL_1:39 for R being non empty multLoopStr, A being non empty Subset of R, L be RightLinearCombination of A ex E be FinSequence of [:the carrier of R, the carrier of R:] st E represents L; theorem :: IDEAL_1:40 for R, S being non empty multLoopStr, F being non empty Subset of R, lc being RightLinearCombination of F, G being non empty Subset of S, P being Function of the carrier of R, the carrier of S, E being FinSequence of [:the carrier of R, the carrier of R:] st P.:F c= G & E represents lc holds ex LC being RightLinearCombination of G st len lc = len LC & for i being set st i in dom LC holds LC.i = (P.(E/.i)`1)*(P.(E/.i)`2); theorem :: IDEAL_1:41 for R being non empty multLoopStr, A being non empty Subset of R, l being LinearCombination of A, n being Nat holds l|Seg n is LinearCombination of A; theorem :: IDEAL_1:42 for R being non empty multLoopStr, A being non empty Subset of R, l being LeftLinearCombination of A, n being Nat holds l|Seg n is LeftLinearCombination of A; theorem :: IDEAL_1:43 for R being non empty multLoopStr, A being non empty Subset of R, l being RightLinearCombination of A, n being Nat holds l|Seg n is RightLinearCombination of A; begin :: Generated ideals definition let L be non empty doubleLoopStr, F be Subset of L; assume F is non empty; func F-Ideal -> Ideal of L means :: IDEAL_1:def 15 F c= it & for I being Ideal of L st F c= I holds it c= I; func F-LeftIdeal -> LeftIdeal of L means :: IDEAL_1:def 16 F c= it & for I being LeftIdeal of L st F c= I holds it c= I; func F-RightIdeal -> RightIdeal of L means :: IDEAL_1:def 17 F c= it & for I being RightIdeal of L st F c= I holds it c= I; end; theorem :: IDEAL_1:44 for L being non empty doubleLoopStr, I being Ideal of L holds I-Ideal = I; theorem :: IDEAL_1:45 for L being non empty doubleLoopStr, I being LeftIdeal of L holds I-LeftIdeal = I; theorem :: IDEAL_1:46 for L being non empty doubleLoopStr, I being RightIdeal of L holds I-RightIdeal = I; definition let L be non empty doubleLoopStr, I be Ideal of L; mode Basis of I -> non empty Subset of L means :: IDEAL_1:def 18 it-Ideal = I; end; theorem :: IDEAL_1:47 for L being add-associative right_zeroed right_complementable distributive (non empty doubleLoopStr) holds {0.L}-Ideal = {0.L}; theorem :: IDEAL_1:48 for L being left_zeroed right_zeroed add-cancelable distributive (non empty doubleLoopStr) holds {0.L}-Ideal = {0.L}; theorem :: IDEAL_1:49 for L being left_zeroed right_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr) holds {0.L}-LeftIdeal = {0.L}; theorem :: IDEAL_1:50 for L being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr) holds {0.L}-RightIdeal = {0.L}; theorem :: IDEAL_1:51 for L being well-unital (non empty doubleLoopStr) holds {1_ L}-Ideal = the carrier of L; theorem :: IDEAL_1:52 for L being right_unital (non empty doubleLoopStr) holds {1_ L}-LeftIdeal = the carrier of L; theorem :: IDEAL_1:53 for L being left_unital (non empty doubleLoopStr) holds {1_ L}-RightIdeal = the carrier of L; theorem :: IDEAL_1:54 for L being non empty doubleLoopStr holds ([#] L)-Ideal = the carrier of L; theorem :: IDEAL_1:55 for L being non empty doubleLoopStr holds ([#] L)-LeftIdeal = the carrier of L; theorem :: IDEAL_1:56 for L being non empty doubleLoopStr holds ([#] L)-RightIdeal = the carrier of L; theorem :: IDEAL_1:57 for L being non empty doubleLoopStr, A, B being non empty Subset of L st A c= B holds A-Ideal c= B-Ideal; theorem :: IDEAL_1:58 for L being non empty doubleLoopStr, A, B being non empty Subset of L st A c= B holds A-LeftIdeal c= B-LeftIdeal; theorem :: IDEAL_1:59 for L being non empty doubleLoopStr, A, B being non empty Subset of L st A c= B holds A-RightIdeal c= B-RightIdeal; theorem :: IDEAL_1:60 for L being add-associative left_zeroed right_zeroed add-cancelable associative distributive well-unital (non empty doubleLoopStr), F being non empty Subset of L, x being set holds x in F-Ideal iff ex f being LinearCombination of F st x = Sum f; theorem :: IDEAL_1:61 for L being add-associative left_zeroed right_zeroed add-cancelable associative distributive well-unital (non empty doubleLoopStr), F being non empty Subset of L, x being set holds x in F-LeftIdeal iff ex f being LeftLinearCombination of F st x = Sum f ; theorem :: IDEAL_1:62 for L being add-associative left_zeroed right_zeroed add-cancelable associative distributive well-unital (non empty doubleLoopStr), F being non empty Subset of L, x being set holds x in F-RightIdeal iff ex f being RightLinearCombination of F st x = Sum f; theorem :: IDEAL_1:63 for R being add-associative left_zeroed right_zeroed add-cancelable well-unital associative commutative distributive (non empty doubleLoopStr), F being non empty Subset of R holds F-Ideal = F-LeftIdeal & F-Ideal = F-RightIdeal; theorem :: IDEAL_1:64 for R being add-associative left_zeroed right_zeroed add-cancelable well-unital associative commutative distributive (non empty doubleLoopStr), a being Element of R holds {a}-Ideal = {a*r where r is Element of R : not contradiction}; theorem :: IDEAL_1:65 for R being Abelian left_zeroed right_zeroed add-cancelable well-unital add-associative associative commutative distributive (non empty doubleLoopStr), a,b being Element of R holds {a,b}-Ideal = {a*r + b*s where r,s is Element of R : not contradiction}; theorem :: IDEAL_1:66 for R being non empty doubleLoopStr, a being Element of R holds a in {a}-Ideal; theorem :: IDEAL_1:67 for R being Abelian left_zeroed right_zeroed right_complementable add-associative associative commutative distributive well-unital (non empty doubleLoopStr), A being non empty Subset of R, a being Element of R holds a in A-Ideal implies {a}-Ideal c= A-Ideal; theorem :: IDEAL_1:68 for R being non empty doubleLoopStr, a,b being Element of R holds a in {a,b}-Ideal & b in {a,b}-Ideal; theorem :: IDEAL_1:69 for R being non empty doubleLoopStr, a,b being Element of R holds {a}-Ideal c= {a,b}-Ideal & {b}-Ideal c= {a,b}-Ideal; begin :: Some Operations on Ideals definition let R be non empty HGrStr, I be Subset of R, a be Element of R; func a*I -> Subset of R equals :: IDEAL_1:def 19 {a*i where i is Element of R : i in I}; end; definition let R be non empty multLoopStr, I be non empty Subset of R, a be Element of R; cluster a*I -> non empty; end; definition let R be distributive (non empty doubleLoopStr), I be add-closed Subset of R, a be Element of R; cluster a*I -> add-closed; end; definition let R be associative (non empty doubleLoopStr), I be right-ideal Subset of R, a be Element of R; cluster a*I -> right-ideal; end; theorem :: IDEAL_1:70 for R being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I being non empty Subset of R holds 0.R*I = {0.R}; theorem :: IDEAL_1:71 for R being left_unital (non empty doubleLoopStr), I being Subset of R holds 1_ R*I = I; definition let R be non empty LoopStr, I,J be Subset of R; func I + J -> Subset of R equals :: IDEAL_1:def 20 {a + b where a,b is Element of R : a in I & b in J}; end; definition let R be non empty LoopStr, I,J be non empty Subset of R; cluster I + J -> non empty; end; definition let R be Abelian (non empty LoopStr), I,J be Subset of R; redefine func I + J; commutativity; end; definition let R be Abelian add-associative (non empty LoopStr), I,J be add-closed Subset of R; cluster I + J -> add-closed; end; definition let R be left-distributive (non empty doubleLoopStr), I,J be right-ideal Subset of R; cluster I + J -> right-ideal; end; definition let R be right-distributive (non empty doubleLoopStr), I,J be left-ideal Subset of R; cluster I + J -> left-ideal; end; theorem :: IDEAL_1:72 for R being add-associative (non empty LoopStr), I,J,K being Subset of R holds I + (J + K) = (I + J) + K; theorem :: IDEAL_1:73 for R being left_zeroed right_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), I,J being right-ideal (non empty Subset of R) holds I c= I + J; theorem :: IDEAL_1:74 for R being left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), I,J being right-ideal (non empty Subset of R) holds J c= I + J; theorem :: IDEAL_1:75 for R being non empty LoopStr, I,J being Subset of R, K being add-closed Subset of R st I c= K & J c= K holds I + J c= K; theorem :: IDEAL_1:76 for R being Abelian left_zeroed right_zeroed add-cancelable well-unital add-associative associative commutative distributive (non empty doubleLoopStr), a,b being Element of R holds {a,b}-Ideal = {a}-Ideal + {b}-Ideal; definition let R be non empty 1-sorted, I,J be Subset of R; func I /\ J -> Subset of R equals :: IDEAL_1:def 21 { x where x is Element of R : x in I & x in J }; end; definition let R be right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I,J be left-ideal (non empty Subset of R); cluster I /\ J -> non empty; end; definition let R be non empty LoopStr, I,J be add-closed Subset of R; cluster I /\ J -> add-closed; end; definition let R be non empty multLoopStr, I,J be left-ideal Subset of R; cluster I /\ J -> left-ideal; end; definition let R be non empty multLoopStr, I,J be right-ideal Subset of R; cluster I /\ J -> right-ideal; end; theorem :: IDEAL_1:77 for R being non empty 1-sorted, I,J being Subset of R holds I /\ J c= I & I /\ J c= J; theorem :: IDEAL_1:78 for R being non empty 1-sorted, I,J,K being Subset of R holds I /\ (J /\ K) = (I /\ J) /\ K; theorem :: IDEAL_1:79 for R being non empty 1-sorted, I,J,K being Subset of R st K c= I & K c= J holds K c= I /\ J; theorem :: IDEAL_1:80 for R being Abelian left_zeroed right_zeroed right_complementable left_unital add-associative left-distributive (non empty doubleLoopStr), I being add-closed left-ideal (non empty Subset of R), J being Subset of R, K being non empty Subset of R st J c= I holds I /\ (J + K) = J + (I /\ K); definition let R be non empty doubleLoopStr, I,J be Subset of R; func I *' J -> Subset of R equals :: IDEAL_1:def 22 { Sum s where s is FinSequence of the carrier of R : for i being Nat st 1 <= i & i <= len s ex a,b being Element of R st s.i = a*b & a in I & b in J}; end; definition let R be non empty doubleLoopStr, I,J be Subset of R; cluster I *' J -> non empty; end; definition let R be commutative (non empty doubleLoopStr), I,J be Subset of R; redefine func I *' J; commutativity; end; definition let R be right_zeroed add-associative (non empty doubleLoopStr), I,J be Subset of R; cluster I *' J -> add-closed; end; definition let R be right_zeroed add-left-cancelable associative left-distributive (non empty doubleLoopStr), I,J be right-ideal Subset of R; cluster I *' J -> right-ideal; end; definition let R be left_zeroed add-right-cancelable associative right-distributive (non empty doubleLoopStr), I,J be left-ideal Subset of R; cluster I *' J -> left-ideal; end; theorem :: IDEAL_1:81 for R being left_zeroed right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I being non empty Subset of R holds {0.R} *' I = {0.R}; theorem :: IDEAL_1:82 for R being left_zeroed right_zeroed add-cancelable distributive (non empty doubleLoopStr), I being add-closed right-ideal (non empty Subset of R), J being add-closed left-ideal (non empty Subset of R) holds I *' J c= I /\ J; theorem :: IDEAL_1:83 for R being Abelian left_zeroed right_zeroed add-cancelable add-associative associative distributive (non empty doubleLoopStr), I,J,K being right-ideal (non empty Subset of R) holds I *' (J + K) = (I *' J) + (I *' K); theorem :: IDEAL_1:84 for R being Abelian left_zeroed right_zeroed add-cancelable add-associative commutative associative distributive (non empty doubleLoopStr), I,J being right-ideal (non empty Subset of R) holds (I + J) *' (I /\ J) c= I *' J; theorem :: IDEAL_1:85 for R being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I,J being add-closed left-ideal (non empty Subset of R) holds (I + J) *' (I /\ J) c= I /\ J; definition let R be non empty LoopStr, I,J be Subset of R; pred I,J are_co-prime means :: IDEAL_1:def 23 I + J = the carrier of R; end; theorem :: IDEAL_1:86 for R being left_zeroed left_unital (non empty doubleLoopStr), I,J being non empty Subset of R st I,J are_co-prime holds I /\ J c= (I + J) *' (I /\ J); theorem :: IDEAL_1:87 for R being Abelian left_zeroed right_zeroed add-cancelable add-associative left_unital commutative associative distributive (non empty doubleLoopStr), I being add-closed left-ideal right-ideal (non empty Subset of R), J being add-closed left-ideal (non empty Subset of R) st I,J are_co-prime holds I *' J = I /\ J; definition let R be non empty HGrStr, I,J be Subset of R; func I % J -> Subset of R equals :: IDEAL_1:def 24 {a where a is Element of R: a*J c= I}; end; definition let R be right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I,J be left-ideal (non empty Subset of R); cluster I % J -> non empty; end; definition let R be right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I,J be add-closed left-ideal (non empty Subset of R); cluster I % J -> add-closed; end; definition let R be right_zeroed add-left-cancelable left-distributive associative commutative (non empty doubleLoopStr), I,J be left-ideal (non empty Subset of R); cluster I % J -> left-ideal; cluster I % J -> right-ideal; end; theorem :: IDEAL_1:88 for R being (non empty multLoopStr), I being right-ideal (non empty Subset of R), J being Subset of R holds I c= I % J; theorem :: IDEAL_1:89 for R being right_zeroed add-left-cancelable left-distributive (non empty doubleLoopStr), I being add-closed left-ideal (non empty Subset of R), J being Subset of R holds (I % J) *' J c= I; theorem :: IDEAL_1:90 for R being left_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), I being add-closed right-ideal (non empty Subset of R), J being Subset of R holds (I % J) *' J c= I; theorem :: IDEAL_1:91 for R being left_zeroed add-right-cancelable right-distributive commutative associative (non empty doubleLoopStr), I being add-closed right-ideal (non empty Subset of R), J,K being Subset of R holds (I % J) % K = I % (J *' K); theorem :: IDEAL_1:92 for R being non empty multLoopStr, I,J,K being Subset of R holds (J /\ K) % I = (J % I) /\ (K % I); theorem :: IDEAL_1:93 for R being left_zeroed right_zeroed add-right-cancelable right-distributive (non empty doubleLoopStr), I being add-closed Subset of R, J,K being right-ideal (non empty Subset of R) holds I % (J + K) = (I % J) /\ (I % K); definition let R be unital (non empty doubleLoopStr), I be Subset of R; func sqrt I -> Subset of R equals :: IDEAL_1:def 25 {a where a is Element of R: ex n being Nat st a|^n in I}; end; definition let R be unital (non empty doubleLoopStr), I be non empty Subset of R; cluster sqrt I -> non empty; end; definition let R be Abelian add-associative left_zeroed right_zeroed commutative associative add-cancelable distributive unital (non empty doubleLoopStr), I be add-closed right-ideal (non empty Subset of R); cluster sqrt I -> add-closed; end; definition let R be unital commutative associative (non empty doubleLoopStr), I be left-ideal (non empty Subset of R); cluster sqrt I -> left-ideal; cluster sqrt I -> right-ideal; end; theorem :: IDEAL_1:94 for R being unital associative (non empty doubleLoopStr), I being non empty Subset of R, a being Element of R holds a in sqrt I iff ex n being Nat st a|^n in sqrt I; theorem :: IDEAL_1:95 for R being left_zeroed right_zeroed add-cancelable distributive unital associative (non empty doubleLoopStr), I being add-closed right-ideal (non empty Subset of R), J being add-closed left-ideal (non empty Subset of R) holds sqrt (I *' J) = sqrt (I /\ J); begin :: Noetherian ideals definition let L be non empty doubleLoopStr, I be Ideal of L; attr I is finitely_generated means :: IDEAL_1:def 26 ex F being non empty finite Subset of L st I = F-Ideal; end; definition let L be non empty doubleLoopStr; cluster finitely_generated Ideal of L; end; definition let L be non empty doubleLoopStr, F be non empty finite Subset of L; cluster F-Ideal -> finitely_generated; end; definition let L be non empty doubleLoopStr; attr L is Noetherian means :: IDEAL_1:def 27 for I being Ideal of L holds I is finitely_generated; end; definition cluster Euclidian Abelian add-associative right_zeroed right_complementable well-unital distributive commutative associative non degenerated (non empty doubleLoopStr); end; definition let L be non empty doubleLoopStr; let I be Ideal of L; attr I is principal means :: IDEAL_1:def 28 ex e being Element of L st I = {e}-Ideal; end; definition let L be non empty doubleLoopStr; attr L is PID means :: IDEAL_1:def 29 for I being Ideal of L holds I is principal; end; theorem :: IDEAL_1:96 for L being non empty doubleLoopStr, F being non empty Subset of L st F <> {0.L} ex x being Element of L st x <> 0.L & x in F; theorem :: IDEAL_1:97 for R being add-associative left_zeroed right_zeroed right_complementable distributive left_unital Euclidian (non empty doubleLoopStr) holds R is PID; theorem :: IDEAL_1:98 for L being non empty doubleLoopStr st L is PID holds L is Noetherian; definition cluster INT.Ring -> Noetherian; end; definition cluster Noetherian Abelian add-associative right_zeroed right_complementable well-unital distributive commutative associative non degenerated (non empty doubleLoopStr); end; theorem :: IDEAL_1:99 :: Lemma_4_5_i_ii: for R being Noetherian add-associative left_zeroed right_zeroed add-cancelable associative distributive well-unital (non empty doubleLoopStr) for B being non empty Subset of R ex C being non empty finite Subset of R st C c= B & C-Ideal = B-Ideal; theorem :: IDEAL_1:100 :: Lemma_4_5_ii_iii: for R being (non empty doubleLoopStr) st for B being non empty Subset of R ex C being non empty finite Subset of R st C c= B & C-Ideal = B-Ideal for a being sequence of R ex m being Nat st a.(m+1) in (rng (a|Segm(m+1)))-Ideal; definition let X, Y be non empty set, f be Function of X, Y, A be non empty Subset of X; cluster f|A -> non empty; end; theorem :: IDEAL_1:101 :: Lemma_4_5_iii_iv: for R being (non empty doubleLoopStr) st for a being sequence of R ex m being Nat st a.(m+1) in (rng (a|Segm(m+1)))-Ideal holds not ex F being Function of NAT, bool (the carrier of R) st (for i being Nat holds F.i is Ideal of R) & (for j,k being Nat st j < k holds F.j c< F.k); theorem :: IDEAL_1:102 :: Lemma_4_5_iv_i: for R being (non empty doubleLoopStr) st not ex F being Function of NAT, bool (the carrier of R) st (for i being Nat holds F.i is Ideal of R) & (for j,k being Nat st j < k holds F.j c< F.k) holds R is Noetherian;

Back to top