Definition:Unitary Module
Jump to navigation
Jump to search
This article is complete as far as it goes, but it could do with expansion, in particular:
Distinguish between a unitary left module and unitary right module
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
Distinguish between a unitary left module and unitary right module
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
To discuss it in more detail, feel free to use the talk page.
If you are able to do this, then when you have done so you can remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage).
Definition
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:
| \((\text {UM} 1)\) | $:$ | \(\displaystyle \forall \lambda \in R: \forall x, y \in G:\) | \(\displaystyle \lambda \circ \paren {x +_G y} = \paren {\lambda \circ x} +_G \paren {\lambda \circ y} \) | |||||
| \((\text {UM} 2)\) | $:$ | \(\displaystyle \forall \lambda, \mu \in R: \forall x \in G:\) | \(\displaystyle \paren {\lambda +_R \mu} \circ x = \paren {\lambda \circ x} +_G \paren {\mu \circ x} \) | |||||
| \((\text {UM} 3)\) | $:$ | \(\displaystyle \forall \lambda, \mu \in R: \forall x \in G:\) | \(\displaystyle \paren {\lambda \times_R \mu} \circ x = \lambda \circ \paren {\mu \circ x} \) | |||||
| \((\text {UM} 4)\) | $:$ | \(\displaystyle \forall x \in G:\) | \(\displaystyle 1_R \circ x = x \) |
Also known as
A unitary module is also known as a unital module. A unitary module over $R$ can also be referred to as a unitary $R$-module.
Also see
- Results about unitary modules can be found here.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 26$. Vector Spaces and Modules
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: Module