Let be a commutative unital ring (often a field) and a -module. Given a bilinear mapping , we say is a -algebra. We usually write only for the tuple .

Remark 1   Many authors and applications insist on as a field, or at least a local ring, or a semisimple ring. This enables to have some notion of dimension or rank.

This definition is a compact method to encode the property that our multiplication is distributive: the multiplication is additive in both variables translates to

Perhaps the most important outcome of these two axioms of an algebra is the opportunity to express polynomial like equations over the algebra. Without the distributive axiom we cannot establish connections between addition and multiplication. Without scalar multiplication we cannot describe coefficients. With these equations we can define certain subalgebras, for example we see both axioms at work in

Proposition 2   Given an algebra , the set

is a submodule of .

Proof. For now let elements of be denoted with to distinguish them from scalars. As a module for all . Then

So .

Also given then for all ,

So .

Finally, given we have

Although this set appears like a reasonable object to define as the center of an algebra, it is usually preferable to produce a subalgebra, not simply a submodule, and for this we need elements that can be regrouped in products associatively, that is, that lie in the nucleus. So the center is commonly defined as

When the algebra has an identity (unity) then we can go further to identify as a subalgebra of by . Then we see this subalgebra is necessarily in the center of . As a converse, given a unital ring (associativity is necessary), the center of the ring forms a commutative unital subring over which is an algebra. In this way unital rings and associative unital algebras are often interchanged.