Definition talk:Module

Left vs. right modules

Hello. Should a distinction be made between left $R$-modules and right $R$-modules? --Jshflynn 18:34, 2 August 2012 (UTC)

I haven't seen this distinction made in any of the literature I've encountered. I can imagine what a right module is but apart from noting that every left module is trivially isomorphic to every right module (and therefore ultimately "the same thing") I can't immediately off the top of my head see why one would bother to design it. --prime mover 19:02, 2 August 2012 (UTC)

Yeah. It's probably the gentlest way to introduce an antihomomorphism (or more accurately an antiautomorphism) to someone but other than that... --Jshflynn 19:16, 2 August 2012 (UTC)

Yes, they are useful and one proves nontrivial things with them (e.g. existence of injectives, also in the commutative case). --barto (talk) (contribs) 12:06, 22 December 2017 (EST)

Rename to "module over ring"?

"Module" is more generally used for anything on which something acts (in this case: a ring action). We have Definition:G-Module, but there is much more, and they are referred to simply as "modules". Should this page be renamed to "R-Module" or "Module over Ring"? --barto (talk) (contribs) 12:06, 22 December 2017 (EST)

There's already an "also known as". --prime mover (talk) 12:31, 22 December 2017 (EST)

Further thought: as long as we don't make "module" a disambiguation page, we can possible transclude "R-Module" and "G-Module" as subpages of a more general "Module" page. If there then proves to be a definition of "module" which is a completely different concept, then we can rethink how we present it. --prime mover (talk) 12:48, 22 December 2017 (EST)

Confused by definition of Module over R

In P. M. Cohn: Basic Algebra: Groups, Rings and Fields, a left $R$-module and a right $R$-module are defined similarly to the definition of a left module over $R$ and a right module over $R$ on $\mathsf{Pr} \infty \mathsf{fWiki}$. The definition of an $R$-module is that it is either a left $R$-module or a right $R$-module but the side is not specified.

On the other hand, Cohn defines an $(R,S)$-bimodule $M$ to be a left $R$-module and a right $S$-module such that:

$\forall \lambda \in R, x \in M, \mu \in S: \lambda \circ \paren {x \circ \mu} = \paren {\lambda \circ x} \circ \mu$

If $R=S$, then $M$ is called an $R$-bimodule. So an $R$-bimodule $M$ is both a left $R$-module and a right $R$-module such that:

$\forall \lambda, \mu \in R, x \in M: \lambda \circ \paren {x \circ \mu} = \paren {\lambda \circ x} \circ \mu$.

The definition of a module over $R$ on $\mathsf{Pr} \infty \mathsf{fWiki}$ is worded to be closer to the definition of an $R$-bimodule than the definition of an $R$-module in Cohn. Yet the usage of the definition module over $R$ in theorems implies that the definition of a module over $R$ is closer to the definition of $R$-module in Cohn.

If the definition on $\mathsf{Pr} \infty \mathsf{fWiki}$ of a module over R is indeed meant to be both a left module over $R$ and a right module over $R$, what, if anything, is the relationship between $\lambda \circ x$ and $x \circ \lambda$. Also is there any relationship between $\lambda \circ \paren {x \circ \mu}$ and $\paren {\lambda \circ x} \circ \mu$. --Leigh.Samphier (talk) 06:34, 12 August 2019 (EDT)