Noncommutative algebra

Noncommutative algebra studies rings whose multiplication need not commute and their one-sided ideals and modules.

A ring is left Noetherian when every ascending chain of left ideals stabilizes, equivalently when every left ideal is finitely generated.

A ring is right Noetherian when every ascending chain of right ideals stabilizes.

The first Weyl algebra is $A_1(k)=k⟨x,y⟩/(yx-xy-1)$, the algebra of polynomial-coefficient differential operators in one variable.

A left primitive ring has a faithful simple left module.

If $V$ is a simple left $R$-module and $D={\mathrm{End}}_R(V{)}^{\mathrm{op}}$, then every prescribed map on a finite $D$-linearly independent subset of $V$ is induced by an element of $R$.

Morita-equivalent rings have equivalent module categories. In particular, $R$ and ${\mathrm{Mat}}_n(R)$ are Morita equivalent.

The Jacobson radical is the intersection of all maximal left ideals, equivalently the intersection of the annihilators of all simple left modules.

An injective module has the extension property for homomorphisms defined on submodules.

A left $R$-module is injective exactly when every homomorphism from a left ideal of $R$ extends to a homomorphism from $R$.

A ring is left Noetherian exactly when every direct sum of injective left modules is injective.

Over an integral domain, a module $M$ is divisible when $rM=M$ for every nonzero $r$. Over a principal ideal domain, divisibility is equivalent to injectivity.

The injective hull of a module $M$ is a minimal injective extension of $M$ in which $M$ is an essential submodule.

Ore localization extends localization to suitable multiplicative subsets of noncommutative rings.

A multiplicative subset $S$ is left Ore when, for every $s\in S$ and $r\in R$, there are $s^{\prime }\in S$ and $r^{\prime }\in R$ such that $s^{\prime }r=r^{\prime }s$.

A proper two-sided ideal $P$ of a noncommutative ring is prime when $AB\subseteq P$ for two-sided ideals $A,B$ implies $A\subseteq P$ or $B\subseteq P$.