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).
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 11.2.2. 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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