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In a commutative ring of prime characteristic $p$, the freshman's dream identity is $(x+y{)}^p=x^p+y^p$. The binomial theorem proves it because $p$ divides every intermediate binomial coefficient $\left(\genfrac{}{}{0ex}{}{p}{k}\right)$, $0<k<p$. Iterating gives $(x+y{)}^{p^m}=x^{p^m}+y^{p^m}$. For example, $(x+y{)}^3=x^3+y^3$ over ${\mathbb{F}}_3$, while over the real numbers the cross terms generally remain. Prime characteristic is essential to this ring-wide statement; an arbitrary positive composite characteristic is insufficient.

An $A$-module is a semisimple module when it is a direct sum of simple modules, equivalently when every submodule has a complementary submodule. The finite-dimensional algebra $A$ is a semisimple algebra when its left regular module ${}_{A}A$ is semisimple.

Let $C_{p^n}=⟨g⟩$, put $N=p^n$, and suppose $\mathrm{char}k=p$. The freshman's dream givesso, with $u=g-1$,The indecomposable modules of a cyclic p-group in characteristic p are preciselyIndeed, a module is a vector space with a nilpotent operator $u$, and its decomposition into Nilpotent Jordan blocks gives these modules. Each $M_r$ is a uniserial module, with unique chainTherefore