Extending an Additive Group Homomorph. to a Ring Homomorph.

[Bashyboy]

Homework Statement

No problem statement.

Homework Equations

The Attempt at a Solution

Suppose that $R$ is a ring and $f : R \to R$ is an additive group homomorphism. Is the following a way of extending $f$ to a ring homomorphism? Let $\varphi : R \to R$ and define $\varphi(r) = f(r)$ if $r \in R - R^\times$, where $R^\times$ is the group of units, and $\varphi(rs) = f(r) f(s)$ if $r,s \in R^\times$...Something about that definition doesn't feel right. Or perhaps $\varphi(a+b) = f(a) + f(b)$ and $\varphi(ab) = f(a) f(b)$...I suspect that that isn't well-defined...I could use some help.

 

[fresh_42]

Homework Statement

No problem statement.

Homework Equations

The Attempt at a Solution

Suppose that $R$ is a ring and $f : R \to R$ is an additive group homomorphism. Is the following a way of extending $f$ to a ring homomorphism? Let $\varphi : R \to R$ and define $\varphi(r) = f(r)$ if $r \in R - R^\times$, where $R^\times$ is the group of units, and $\varphi(rs) = f(r) f(s)$ if $r,s \in R^\times$...Something about that definition doesn't feel right. Or perhaps $\varphi(a+b) = f(a) + f(b)$ and $\varphi(ab) = f(a) f(b)$...I suspect that that isn't well-defined...I could use some help.

There is given a group homomorphism $f : R^+ \longrightarrow R^+$. This naturally extends to a ring homomorphism $\bar{f}: \otimes_\mathbb{Z} R^+ \longrightarrow \otimes_\mathbb{Z} R^+$. To extend $f$ to a ring homomorphism $\varphi : R \longrightarrow R$ we can ask, whether there is an ideal $\mathcal{I}$ in $\otimes_\mathbb{Z} R^+$ such that $\otimes_\mathbb{Z} R^+ / \mathcal{I} \cong R$. If $\pi : \otimes_\mathbb{Z} R^+ \twoheadrightarrow R$ denotes the according projection, then $\varphi$ given by $\varphi \circ \pi = \pi \circ \bar{f}$ should do the job.

The crucial point is to transport the multiplicative rules form $R$ into an ideal of $\otimes_\mathbb{Z} R^+$. I cannot see how units help here. They might not reflect enough of these rules, as they are notoriously exceptional in a ring and don't match very well with the given additive structure.