A simple module (also called an irreducible module) is a fundamental object in module theory, a branch of abstract algebra. Let $R$ be a ring (with identity) and $M$ a left (or right) $R$-module. The module $M$ is called simple if its only submodules are the zero submodule ${0}$ and $M$ itself. Equivalently, $M$ has no proper non‑zero submodules.
Formal Definition
Let $R$ be a ring and $M$ a left $R$-module. $M$ is simple ⇔ $$ \forall, N \leq M,; (N = 0 \text{ or } N = M). $$
The definition for right modules is analogous.
Basic Properties
- Homomorphisms: Any non‑zero homomorphism between simple modules is injective, and because the image is a non‑zero submodule, it must be surjective; thus any non‑zero homomorphism between simple modules is an isomorphism (Schur’s Lemma).
- Composition Series: A module that possesses a finite composition series has simple factors; the Jordan–Hölder theorem guarantees that the multiset of these simple factors is uniquely determined up to order.
- Semisimple Modules: A module is semisimple (or completely reducible) if it is a direct sum of simple modules. Over a semisimple ring, every module is semisimple.
- Endomorphism Ring: For a simple left $R$-module $M$, the ring $\operatorname{End}_R(M)$ is a division ring (by Schur’s Lemma).
Examples
- Vector Spaces over Division Rings: If $D$ is a division ring, any one‑dimensional left $D$-vector space $D$ (viewed as a left $D$-module) is simple, because its only subspaces are ${0}$ and itself.
- Modules over Principal Ideal Domains: For a PID $R$, the quotient $R/(p)$ where $p$ is a prime element is a simple $R$-module.
- Group Representations: Over a field $k$, an irreducible representation of a group $G$ corresponds to a simple $kG$-module, where $kG$ is the group algebra.
- Simple Rings as Modules: If $R$ is a simple ring (i.e., it has no non‑trivial two‑sided ideals), then $R$ regarded as a left (or right) module over itself is simple.
Classification in Specific Settings
- Artinian Rings: Over a left Artinian ring, every simple module is isomorphic to a quotient $R/e$ where $e$ is a primitive idempotent.
- Semisimple Rings (Wedderburn–Artin Theorem): Every simple left module over a semisimple ring $R$ is isomorphic to a minimal left ideal of $R$, and $R$ decomposes as a finite direct product of matrix rings over division rings.
Related Concepts
- Irreducible Representation: In representation theory, an irreducible representation of a group or algebra is precisely a simple module over the corresponding group algebra or algebra.
- Composition Factor: Simple modules that appear as factors in a composition series of a module.
- Radical and Socle: The radical of a module is the intersection of all maximal submodules; the socle is the sum of all simple submodules.
References
- S. Lang, Algebra, Springer, 2002.
- I. N. Herstein, Topics in Algebra, Wiley, 1975.
- J. E. Humphreys, Linear Algebraic Groups, Springer, 1975. (for representation‑theoretic context)