Definition:Algebra over Ring

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Let $R$ be a commutative ring.

An algebra over $R$ is an ordered pair $\struct {A, *}$ where:

$A$ is an $R$-module
$*: A^2 \to A$ is an $R$-bilinear mapping

Commutativity of the Ring

Because the definition of bilinear mapping is intricate in the case of non-commutative rings, it is important that $R$ be commutative.

It does not necessarily have to be a ring with unity.

Also defined as

Sources who only deal with rings with unity often define an algebra as one whose underlying module is unital.

Especially in commutative algebra, an algebra over a commutative ring with unity $R$ is often defined as a unital associative commutative algebra.

Also see

It can be considered to be a generalization of an algebra over a field in which:

the vector space is replaced by a module
the field is replaced by a commutative ring.
  • Results about algebras can be found here.