Journal of Formalized Mathematics
Volume 4, 1992
University of Bialystok
Copyright (c) 1992 Association of Mizar Users

## Opposite Rings, Modules and Their Morphisms

Michal Muzalewski
Warsaw University, Bialystok

### Summary.

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 $K$.

#### MML Identifier: MOD_4

The terminology and notation used in this paper have been introduced in the following articles [4] [12] [13] [2] [3] [1] [11] [6] [7] [9] [5] [10]

#### Contents (PDF format)

1. Opposite functions
2. Opposite rings
3. Opposite modules
4. Morphisms of rings
5. Opposite morphisms to morphisms of rings
6. Morphisms of groups
7. Semilinear morphisms

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