## 11.2 Noncommutative algebras

Let $k$ be a field. In this chapter an algebra $A$ over $k$ is a possibly noncommutative ring $A$ together with a ring map $k \to A$ such that $k$ maps into the center of $A$ and such that $1$ maps to an identity element of $A$. An $A$-module is a right $A$-module such that the identity of $A$ acts as the identity.

Definition 11.2.1. Let $A$ be a $k$-algebra. We say $A$ is finite if $\dim _ k(A) < \infty$. In this case we write $[A : k] = \dim _ k(A)$.

Definition 11.2.2. A skew field is a possibly noncommutative ring with an identity element $1$, with $1 \not= 0$, in which every nonzero element has a multiplicative inverse.

A skew field is a $k$-algebra for some $k$ (e.g., for the prime field contained in it). We will use below that any module over a skew field is free because a maximal linearly independent set of vectors forms a basis and exists by Zorn's lemma.

Definition 11.2.3. Let $A$ be a $k$-algebra. We say an $A$-module $M$ is simple if it is nonzero and the only $A$-submodules are $0$ and $M$. We say $A$ is simple if the only two-sided ideals of $A$ are $0$ and $A$.

Definition 11.2.4. A $k$-algebra $A$ is central if the center of $A$ is the image of $k \to A$.

Definition 11.2.5. Given a $k$-algebra $A$ we denote $A^{op}$ the $k$-algebra we get by reversing the order of multiplication in $A$. This is called the opposite algebra.

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