Volume 4, 1992

University of Bialystok

Copyright (c) 1992 Association of Mizar Users

**Michal Muzalewski**- Warsaw University, Bialystok

- Let $\Bbb K = \langle S; K, 0, 1, +, \cdot \rangle$ be a ring. The structure ${}^{\rm op}\Bbb K = \langle S; K, 0, 1, +, \bullet \rangle$ is called anti-ring, if $\alpha \bullet \beta = \beta \cdot \alpha$ for elements $\alpha$, $\beta$ of $K$ [8, pages 5-7]. It is easily seen that ${}^{\rm op}\Bbb K$ is also a ring. If $V$ is a left module over $\Bbb K$, then $V$ is a right module over ${}^{\rm op}\Bbb K$. If $W$ is a right module over $\Bbb K$, then $W$ is a left module over ${}^{\rm op}\Bbb K$. Let $K, L$ be rings. A morphism $J: K \longrightarrow L$ is called anti-homomorphism, if $J(\alpha\cdot\beta) = J(\beta)\cdot J(\alpha)$ for elements $\alpha$, $\beta$ of $K$. If $J:K \longrightarrow L$ is a homomorphism, then $J:K \longrightarrow {}^{\rm op}L$ is an anti-homomorphism. Let $K, L$ be rings, $V, W$ left modules over $K, L$ respectively and $J:K \longrightarrow L$ an anti-monomorphism. A map $f:V \longrightarrow W$ is called $J$ - semilinear, if $f(x+y) = f(x)+f(y)$ and $f(\alpha\cdot x) = J(\alpha)\cdot f(x)$ for vectors $x, y$ of $V$ and a scalar $\alpha$ of $K$.

- Opposite functions
- Opposite rings
- Opposite modules
- Morphisms of rings
- Opposite morphisms to morphisms of rings
- Morphisms of groups
- Semilinear morphisms

- [1]
Czeslaw Bylinski.
Binary operations.
*Journal of Formalized Mathematics*, 1, 1989. - [2]
Czeslaw Bylinski.
Functions and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989. - [3]
Czeslaw Bylinski.
Functions from a set to a set.
*Journal of Formalized Mathematics*, 1, 1989. - [4]
Czeslaw Bylinski.
Some basic properties of sets.
*Journal of Formalized Mathematics*, 1, 1989. - [5]
Czeslaw Bylinski.
The modification of a function by a function and the iteration of the composition of a function.
*Journal of Formalized Mathematics*, 2, 1990. - [6]
Eugeniusz Kusak, Wojciech Leonczuk, and Michal Muzalewski.
Abelian groups, fields and vector spaces.
*Journal of Formalized Mathematics*, 1, 1989. - [7]
Michal Muzalewski.
Construction of rings and left-, right-, and bi-modules over a ring.
*Journal of Formalized Mathematics*, 2, 1990. - [8] Michal Muzalewski. \em Foundations of Metric-Affine Geometry. Dzial Wydawnictw Filii UW w Bialymstoku, Filia UW w Bialymstoku, 1990.
- [9]
Michal Muzalewski.
Categories of groups.
*Journal of Formalized Mathematics*, 3, 1991. - [10]
Michal Muzalewski.
Category of rings.
*Journal of Formalized Mathematics*, 3, 1991. - [11]
Wojciech A. Trybulec.
Vectors in real linear space.
*Journal of Formalized Mathematics*, 1, 1989. - [12]
Zinaida Trybulec.
Properties of subsets.
*Journal of Formalized Mathematics*, 1, 1989. - [13]
Edmund Woronowicz.
Relations and their basic properties.
*Journal of Formalized Mathematics*, 1, 1989.

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