Associative Unital Algebra

An associative unital algebra over a Commutative Ring $R$ is a Ring $A$ that is also an R-Module where the two additions coincide, and scalar multiplication satisfies

• $r\cdot(xy)=(r\cdot x)y=x(r\cdot y)$.

Equivalently, an associative unital algebra is a Ring $A$ together with a Ring Homomorphism $\eta:R\to Z(A)$ to the Center of $A$.

More abstractly, an unital associative algebra is a Monoid Object in R-Mod, equipped with the Tensor Product of Modules.

Over Non-Commutative Rings

We can generalize associative algebras by passing to $(R,R)$ Bimodules. Explicitly, an associative algebra over a Ring $R$ is an $(R,R)$-bimodule that is also a Ring, such that scalar multiplication satisfies:

• $r\cdot(xy)=(r\cdot x)y$
• $(xy)\cdot r=x(y\cdot r)$

Moreover, we don't use negatives, so it is natural to consider associative unital algebras over a Semiring $R$.

Question

Is an associative unital algebra a Monad in the Double Category of Bimodules?

Examples

• Every Ring can be considered as a $\mathbb{Z}$-algebra.
• A Ring of Matrices with coefficients in $R$ forms an $R$ algebra under matrix addition and multiplication.