2. Modules

2.1 Definition. Let R be a commutative ring. An Abelian group (A, +) together with a “scalar” multiplication ·: R × A → A by elements of R is called an Rmodule, if the following conditions hold for all r, s ∈ R and a, b ∈ A:

(r + s)a = ra + sa (rs)a = r(sa) r(a + b) = ra + rb In addition, we will always assume that modules are unitary, which means

that multiplication by 1 ∈ R is the identity map on A, i.e., for all a ∈ A, 1_{R}a = a.

2.2 Definition. Given an R-module A, the annihilator of a subset B of A is Ann_{R}(B) = {r ∈ R | ∀b ∈ B rb = 0}.

Clearly, Ann_{R}(B) is an ideal of R for any B. Of particular importance is the annihilator of the whole module. If Ann_{R}(A) = I for an R-module A then A can be regarded as an R/I-module by defining (r + I)a:= ra.

2.3 Definition. An R-module A is called faithful if Ann_{R}(A) = (0).

2.4 Definition. A (possibly non-commutative) ring A is called an R-algebra, if (A, +) is an R-module and the multiplication of the ring A interacts peacefully with the scalar multiplication as follows: for all r ∈ R and a, b ∈ A

r(ab) = (ra)b = a(rb).

Examples of R-algebras include the polynomial ring R[x] and the ring M_{n}(R) of n × n matrices with entries in R. More generally, every ring T of which R is a subring is an R-algebra (by restriction of the ring-multiplication on T to ·: R × T → T).

Because of rb = (r1_{A})b, an R-algebra A is faithful if and only if Ann(1_{A}) = (0), and therefore every faithful R-algebra A admits an embedding of R by r ↦ r1_{A}. If we call R̃ the copy of R thus embedded in A, then, again by rb = (r1_{A})b, the scalar multiplication of the R-algebra A is just the restriction of the ring multiplication of A to R̃ × A.