Does an $R$ algebra always contain $R$?

Question

In an associative algebra with unit over a commutative ring $R$ it's true that $R$ is inside the algebra? And, is $1$ is the unit in the algebra, is this inclusion $R\cdot1$?

Answer 1

If $A$ is an (associative) $R$-algebra with unit, then there is a canonical ring morphism $R\to A$, sending $x\in R$ to $x\cdot 1_A$, where we use the $R$-module structure of $A$. But in general this morphism is not injective, so we cannot really say that $R$ "is inside" $A$.

For instance, $\mathbb{Z}/2\mathbb{Z}$ is a unitary $\mathbb{Z}$-algebra, but the canonical morphism $\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$ is of course not injective.

Note that we can also directly define associative unitary $R$-algebras as being rings $A$ with a ring morphism $R\to Z(A)$, where $Z(A)$ is the center of $A$.