:: Quotient Rings
:: by Artur Korni{\l}owicz
::
:: Received December 7, 2005
:: Copyright (c) 2005 Association of Mizar Users
theorem Th1: :: RING_1:1
theorem Th2: :: RING_1:2
theorem Th3: :: RING_1:3
:: deftheorem Def1 defines quasi-prime RING_1:def 1 :
:: deftheorem Def2 defines prime RING_1:def 2 :
:: deftheorem Def3 defines quasi-maximal RING_1:def 3 :
:: deftheorem Def4 defines maximal RING_1:def 4 :
theorem :: RING_1:4
Lm1:
for R being Ring
for I being Ideal of R ex E being Equivalence_Relation of the carrier of R st
for x, y being set holds
( [x,y] in E iff ( x in the carrier of R & y in the carrier of R & ex P, Q being Element of R st
( P = x & Q = y & P - Q in I ) ) )
definition
let R be
Ring;
let I be
Ideal of
R;
func EqRel R,
I -> Relation of
R means :
Def5:
:: RING_1:def 5
for
a,
b being
Element of
R holds
(
[a,b] in it iff
a - b in I );
existence
ex b1 being Relation of R st
for a, b being Element of R holds
( [a,b] in b1 iff a - b in I )
uniqueness
for b1, b2 being Relation of R st ( for a, b being Element of R holds
( [a,b] in b1 iff a - b in I ) ) & ( for a, b being Element of R holds
( [a,b] in b2 iff a - b in I ) ) holds
b1 = b2
end;
:: deftheorem Def5 defines EqRel RING_1:def 5 :
theorem Th5: :: RING_1:5
theorem Th6: :: RING_1:6
theorem Th7: :: RING_1:7
theorem :: RING_1:8
theorem Th9: :: RING_1:9
theorem :: RING_1:10
definition
let R be
Ring;
let I be
Ideal of
R;
func QuotientRing R,
I -> strict doubleLoopStr means :
Def6:
:: RING_1:def 6
( the
carrier of
it = Class (EqRel R,I) &
1. it = Class (EqRel R,I),
(1. R) &
0. it = Class (EqRel R,I),
(0. R) & ( for
x,
y being
Element of
it ex
a,
b being
Element of
R st
(
x = Class (EqRel R,I),
a &
y = Class (EqRel R,I),
b & the
addF of
it . x,
y = Class (EqRel R,I),
(a + b) ) ) & ( for
x,
y being
Element of
it ex
a,
b being
Element of
R st
(
x = Class (EqRel R,I),
a &
y = Class (EqRel R,I),
b & the
multF of
it . x,
y = Class (EqRel R,I),
(a * b) ) ) );
existence
ex b1 being strict doubleLoopStr st
( the carrier of b1 = Class (EqRel R,I) & 1. b1 = Class (EqRel R,I),(1. R) & 0. b1 = Class (EqRel R,I),(0. R) & ( for x, y being Element of b1 ex a, b being Element of R st
( x = Class (EqRel R,I),a & y = Class (EqRel R,I),b & the addF of b1 . x,y = Class (EqRel R,I),(a + b) ) ) & ( for x, y being Element of b1 ex a, b being Element of R st
( x = Class (EqRel R,I),a & y = Class (EqRel R,I),b & the multF of b1 . x,y = Class (EqRel R,I),(a * b) ) ) )
uniqueness
for b1, b2 being strict doubleLoopStr st the carrier of b1 = Class (EqRel R,I) & 1. b1 = Class (EqRel R,I),(1. R) & 0. b1 = Class (EqRel R,I),(0. R) & ( for x, y being Element of b1 ex a, b being Element of R st
( x = Class (EqRel R,I),a & y = Class (EqRel R,I),b & the addF of b1 . x,y = Class (EqRel R,I),(a + b) ) ) & ( for x, y being Element of b1 ex a, b being Element of R st
( x = Class (EqRel R,I),a & y = Class (EqRel R,I),b & the multF of b1 . x,y = Class (EqRel R,I),(a * b) ) ) & the carrier of b2 = Class (EqRel R,I) & 1. b2 = Class (EqRel R,I),(1. R) & 0. b2 = Class (EqRel R,I),(0. R) & ( for x, y being Element of b2 ex a, b being Element of R st
( x = Class (EqRel R,I),a & y = Class (EqRel R,I),b & the addF of b2 . x,y = Class (EqRel R,I),(a + b) ) ) & ( for x, y being Element of b2 ex a, b being Element of R st
( x = Class (EqRel R,I),a & y = Class (EqRel R,I),b & the multF of b2 . x,y = Class (EqRel R,I),(a * b) ) ) holds
b1 = b2
end;
:: deftheorem Def6 defines QuotientRing RING_1:def 6 :
for
R being
Ring for
I being
Ideal of
R for
b3 being
strict doubleLoopStr holds
(
b3 = QuotientRing R,
I iff ( the
carrier of
b3 = Class (EqRel R,I) &
1. b3 = Class (EqRel R,I),
(1. R) &
0. b3 = Class (EqRel R,I),
(0. R) & ( for
x,
y being
Element of
b3 ex
a,
b being
Element of
R st
(
x = Class (EqRel R,I),
a &
y = Class (EqRel R,I),
b & the
addF of
b3 . x,
y = Class (EqRel R,I),
(a + b) ) ) & ( for
x,
y being
Element of
b3 ex
a,
b being
Element of
R st
(
x = Class (EqRel R,I),
a &
y = Class (EqRel R,I),
b & the
multF of
b3 . x,
y = Class (EqRel R,I),
(a * b) ) ) ) );
theorem Th11: :: RING_1:11
theorem Th12: :: RING_1:12
theorem Th13: :: RING_1:13
theorem Th14: :: RING_1:14
Lm2:
now
let R be
Ring;
:: thesis: for I being Ideal of R
for e being Element of (R / I) st e = Class (EqRel R,I),(1_ R) holds
for h being Element of (R / I) holds
( h * e = h & e * h = h )let I be
Ideal of
R;
:: thesis: for e being Element of (R / I) st e = Class (EqRel R,I),(1_ R) holds
for h being Element of (R / I) holds
( h * e = h & e * h = h )set E =
EqRel R,
I;
let e be
Element of
(R / I);
:: thesis: ( e = Class (EqRel R,I),(1_ R) implies for h being Element of (R / I) holds
( h * e = h & e * h = h ) )assume A1:
e = Class (EqRel R,I),
(1_ R)
;
:: thesis: for h being Element of (R / I) holds
( h * e = h & e * h = h )let h be
Element of
(R / I);
:: thesis: ( h * e = h & e * h = h )consider a being
Element of
R such that A2:
e = Class (EqRel R,I),
a
by Th11;
consider b being
Element of
R such that A3:
h = Class (EqRel R,I),
b
by Th11;
A4:
a - (1_ R) in I
by A1, A2, Th6;
then A5:
b * (a - (1_ R)) in I
by IDEAL_1:def 2;
A6:
b * (a - (1_ R)) =
(b * a) - (b * (1_ R))
by VECTSP_1:43
.=
(b * a) - b
by VECTSP_1:def 13
;
thus h * e =
Class (EqRel R,I),
(b * a)
by A2, A3, Th14
.=
h
by A3, A5, A6, Th6
;
:: thesis: e * h = hA7:
(a - (1_ R)) * b in I
by A4, IDEAL_1:def 3;
A8:
(a - (1_ R)) * b =
(a * b) - ((1_ R) * b)
by VECTSP_1:45
.=
(a * b) - b
by VECTSP_1:def 19
;
thus e * h =
Class (EqRel R,I),
(a * b)
by A2, A3, Th14
.=
h
by A3, A7, A8, Th6
;
:: thesis: verum
end;
theorem :: RING_1:15
theorem Th16: :: RING_1:16
theorem Th17: :: RING_1:17
theorem :: RING_1:18
theorem Th19: :: RING_1:19
theorem Th20: :: RING_1:20
theorem :: RING_1:21