Let $A$ be an associative ring (not necessarily with unit) and $P$ a two-sided $S$-module. In addition, assume that $A$ and $P$ are modules over a commutative ring $S$ and that all operations of $A$ on $A$ and $P$ are $S$-homomorphisms. Defining the cohomology groups of the $S$-algebra $A$ with coefficients in $P$ by the usual cochain formulae, the author is primarily interested in the ring structure of $H^*(A,A)$ and the module structure over $H^*(A,A)$ of $H^*(A,P)$ given by the usual cup product multiplication. Among other things, it is shown that $H^*(A,A)$ is a commutative ring in the sense of graded rings. A bracket product $[ , ]$ is introduced in $H^*(A,A)$ under which $H^*(A,A)$ becomes a graded Lie ring with $[H^m(A,A),H^n(A,A)]⊂ H^m+n-1(A,A)$ and such that the bracket operation of $H^1(A,A)$ into itself is the ordinary Poisson bracket of derivations of $A$ into itself. Various other properties of this operation are derived and its role in the author's theory of deformations of algebras indicated.
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