# Extending an Additive Group Homomorph. to a Ring Homomorph.

Bashyboy

## Homework Statement

No problem statement.

## 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.

Mentor
2022 Award

## Homework Statement

No problem statement.

## 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.

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