[[R-algebra]]
# $R$-monoid

Let $R$ be a [[commutative ring]].
An **$R$-monoid** $T$ is a [[Monoid object|monoid]] in the category [[Category of modules over a commutative ring|$\lMod{R}$]].
More concretely, an **$R$-monoid** $T$ can be viewed in two equivalent ways: #m/def/calg

1. As an [[R-algebra]] $T$ which is unital and associative;
2. As a [[ring]] $T$ equipped with a homomorphism $R \to \opn Z(T)$ into its [[Centre of a rng|centre]].

This is of course a strenthening of [[R-semigroup]].
It follows every [[ring]] is a [[Integers]]-monoid in a unique way.

## See also

- [[R-comonoid]]

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#state/tidy | #lang/en | #SemBr