Algebras smaller than their underlying ring.

Question

Algebras are defined with respect to an underlying ring. Now it makes sense that the underlying ring is smaller. But I was wondering if it is possible to have an underlying ring which is larger. In this case the homomorphism from the ring $R$ to the algebra $A$ $$\alpha : R \rightarrow A$$ will be a very weird one. Is there any rule for stopping the existence of such a weird homomorphism? Or do such algebras exist?

Answer 1

Take the coefficient ring to be $\mathbb Z$ and the algebra to be $\mathbb Z/n\mathbb Z$. Any finite ring will also do as an example because any ring is a $\mathbb Z$-algebra.

Answer 2

You may consider $R=k[X]$ and $A=k$ with $\alpha(X)=c$ for some constant $c\in k$. Though this is not particularly interesting.