Algebra over a field
Associative algebras over rings

A ring A is always an associative algebra over its center, and over the integers. A classical example of an algebra over its center is the split-biquaternion algebra, which is isomorphic to H × H {\displaystyle \mathbb {H} \times \mathbb {H} }, the direct product of two quaternion algebras. The center of that ring is R × R {\displaystyle \mathbb {R} \times \mathbb {R} }, and hence it has the structure of an algebra over its center, which is not a field. Note that the split-biquaternion algebra is also naturally an 8-dimensional R {\displaystyle \mathbb {R} } -algebra.

In commutative algebra, if A is a commutative ring, then any unital ring homomorphism R → A {\displaystyle R\to A} defines an R -module structure on A, and this is what is known as the R -algebra structure. So a ring comes with a natural Z {\displaystyle \mathbb {Z} } -module structure, since one can take the unique homomorphism Z → A {\displaystyle \mathbb {Z} \to A}. On the other hand, not all rings can be given the structure of an algebra over a field (for example the integers). See Field with one element for a description of an attempt to give to every ring a structure that behaves like an algebra over a field.
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