This chapter develops a number of other concepts concerning rings. These concepts will play important roles later in the text, and we prefer to discuss them now, so as to avoid too many interruptions of the flow of subsequent discussions.

Algebras

Let R be a ring. An R-algebra (or algebra overR) is a ring E, together with a ring homomorphism τ: R → E. Usually, the map τ will be clear from context, as in the following examples.

Example 17.1. If E is a ring that contains R as a subring, then E is an R-algebra, where the associated map τ: R → E is just the inclusion map.

Example 17.2. Let E1, …, En be R-algebras, with associated maps τi: R → Ei, for i = 1, …, n. Then the direct product ring E:= E1 × … × En is naturally viewed as an R-algebra, via the map τ that sends a ∈ R, to (τ1(a), …, τn(a)) ∈ E.

Example 17.3. Let E be an R-algebra, with associated map τ: R → E, and let I be an ideal of E. Consider the quotient ring E/I. If ρ is the natural map from E onto E/I, then the homomorphism ρ ∘ τ makes E/I into an R-algebra, called the quotient algebra ofEmoduloI.