Duality Concept: Submodule and Quotient

Question

Let $R$ be a ring with $1$ and $M$ be an $R$-module. With nice enough conditions, we can define

• $\operatorname{Rad}(M) = \text{(unique) smallest submodule $N$ of $M$ such that $M/N$ is semisimple}$
• $\operatorname{Soc}(M) = \text{(unique) largest semisimple submodule $N$ of $M$}$

So $\operatorname{Rad}(M)$ is dual to $\operatorname{Soc}(M)$.

My question: What does it mean to be dual in this case? These conditions do feel dual but how do we formalize this duality notion?

Answer 1

Here's a somewhat late answer, in case you're still wondering.

TL;DR: When dealing with the poset of submodules of $M$, we have that intersections of submodules are dual to sums of submodules, and maximal submodules are dual to simple submodules.

Longer answer:

The core idea here is that the radical of a module $M$ is not dual to the socle of $M$ in the category $\operatorname{Mod}R$; instead, these concepts are dual in the poset category of submodules of $M$. Let $X$ denote the poset of submodules of $M$ ordered by inclusion, and let $\mathcal{C}_X$ denote its poset category.

To see how the radical and socle are dual, recall that the radical of $M$ is the intersection of all maximal submodules of $M$. We can phrase this in categorical terms:

• The intersection of a set $\{A_i\}_{i\in I}$ of submodules is the largest module which is contained in every $A_i$. In other words, it is the infimum of $\{A_i\}_{i\in I}$ in the poset $X$, which we may view as the limit (or categorical product) of $\{A_i\}_{i\in I}$ in the category $\mathcal{C}_X$.
• A maximal submodule of $M$ is an element $A$ of $X$ such that $A \ne M$ and $A$ is not (strictly) smaller than any element of $X$ except $M$. Equivalently, it is an object $A$ of $\mathcal{C}_X$ which is not the terminal object $M$, and such that if $f: A \to T$ is a morphism in $\mathcal{C}_X$, then either $T = A$ or $T$ is the terminal object $M$. I do not know if there is an established term for such an object, but I will call it a penultimate object.

Thus the radical of $M$ is the product of all penultimate objects in $\mathcal{C}_X$. We can now find the dual concept in the category $\mathcal{C}_X$: the "co-radical" of $M$ is the coproduct of all co-penultimate objects in $\mathcal{C}_X$. Let's unpack what this means:

• If $\{A_i\}_{i \in I}$ is a set of objects in $\mathcal{C}_X$, then their coproduct is the supremum of $\{A_i\}_{i \in I}$ in $X$. But the supremum of a set of submodules of $M$ is just their sum.
• A co-penultimate object in $\mathcal{C}_X$ is a non-initial object $A$ such that if $f: I \to A$ is a morphism in $\mathcal{C}_X$, then either $I = A$ or $I$ is the initial object. But the initial object of $\mathcal{C}_X$ is the zero submodule of $M$, so a co-penultimate object of $\mathcal{C}_X$ is just a nonzero submodule of $M$ with no nontrivial submodules; that is, it is a simple submodule.

Thus the "co-radical" of $M$ is the sum of all its simple submodules. But this is the socle of $M$! So we see that the radical and socle of $M$ are dual in the poset category $\mathcal{C}_X$.