# Category:Unitary Modules

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This category contains results about Unitary Modules.

Definitions specific to this category can be found in Definitions/Unitary Modules.

Let $\struct {R, +_R, \times_R}$ be a ring with unity whose unity is $1_R$.

Let $\struct {G, +_G}$ be an abelian group.

A unitary module over $R$ is an $R$-algebraic structure with one operation $\struct {G, +_G, \circ}_R$ which satisfies the unitary module axioms:

\((\text {UM} 1)\) | $:$ | \(\displaystyle \forall \lambda \in R: \forall x, y \in G:\) | \(\displaystyle \lambda \circ \paren {x +_G y} = \paren {\lambda \circ x} +_G \paren {\lambda \circ y} \) | |||||

\((\text {UM} 2)\) | $:$ | \(\displaystyle \forall \lambda, \mu \in R: \forall x \in G:\) | \(\displaystyle \paren {\lambda +_R \mu} \circ x = \paren {\lambda \circ x} +_G \paren {\mu \circ x} \) | |||||

\((\text {UM} 3)\) | $:$ | \(\displaystyle \forall \lambda, \mu \in R: \forall x \in G:\) | \(\displaystyle \paren {\lambda \times_R \mu} \circ x = \lambda \circ \paren {\mu \circ x} \) | |||||

\((\text {UM} 4)\) | $:$ | \(\displaystyle \forall x \in G:\) | \(\displaystyle 1_R \circ x = x \) |

## Subcategories

This category has the following 2 subcategories, out of 2 total.

### E

### F

## Pages in category "Unitary Modules"

The following 11 pages are in this category, out of 11 total.