The notion of R-algebra generalizes that of an associative algebra: if K is a field, then any associative algebra over K is a K_algebra and vice_versa. Every R_algebra is also an R-module in an obvious manner.
Any ring S can be considered as an algebra over its center R.
Any ring S can be considered as a Z-algebra in a unique way.
A ring is a generalization of the integers, which itself is an example of a ring.
Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutativerings.
Since any ring is both a left and right module over itself, it is possible to construct the tensor product of R over a ring S with another ring T to get another ring provided S is a central subring of R and T.
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