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 $A$.
Sign rules. In this chapter we will work with graded algebras and graded modules often equipped with differentials. The sign rules on underlying complexes will always be (compatible with) those introduced in More on Algebra, Section 15.72. This will occasionally cause the multiplicative structure to be twisted in unexpected ways especially when considering left modules or the relationship between left and right modules.
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