In this chapter we hold on to the convention that ring means commutative ring with $1$. If $R$ is a ring, then an $R$-algebra $A$ will be an $R$-module $A$ endowed with an $R$-bilinear map $A \times A \to A$ (multiplication) such that multiplication is associative and has a unit. In other words, these are unital associative $R$-algebras such that the structure map $R \to A$ maps into the center of $A$.
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