Basic Properties Regarding Ring Homomorphisms

# Basic Properties Regarding Ring Homomorphisms

Recall from the Ring Homomorphisms page that if $(R, +_1, *_1)$ and $(S, +_2, *_2)$ are rings with multiplicative identities $1_R$ and $1_S$ respectively, then a ring homomorphism between them is a function $\phi : R \to S$ such that for all $a, b \in R$ we have that:

• 1) $\phi (a +_1 b) = \phi (a) +_2 \phi (b)$.
• 2) $\phi (a *_1 b) = \phi (a) *_2 \phi (b)$.
• 3) $\phi (1_R) = 1_S$

We now state some basic properties of ring homomorphisms.

 Theorem 1: Let $(R, +_1, *_1)$ and $(S, +_2, *_2)$ be rings with additive identities $0_R$ and $0_S$ respectively. If $R$ and $S$ are homomorphic with homomorphism $\phi$ then $\phi (0_R) = 0_S$.
• Proof: We have that:
(1)
\begin{align} \quad \phi (0_R) = \phi(0_R + 0_R) = \phi (0_R) + \phi(0_R) \end{align}
• Therefore $\phi (0_R) = 0_S$. $\blacksquare$
 Lemma 2: Let $(R, +_1, *_1)$ and $(S, +_2, *_2)$ be rings with additive identities $0_R$ and $0_S$ respectively and multiplicative identities $1_R$ and $1_S$ respectively. If $R$ and $S$ are homomorphic with homomorphism $\phi$ then $\phi (-1_R) = -1_S$.
• Proof: By the previous theorem we have that:
(2)
\begin{align} \quad 0_S = \phi (0_R) = \phi (1_R - 1_R) = \phi (1_R) + \phi(-1_R) = 1_S + \phi(-1_R) \end{align}
• So $\phi (-1_R) = -1_S$. $\blacksquare$
 Theorem 3: Let $(R, +_1, *_1)$ and $(S, +_2, *_2)$ be rings with additive identities $0_R$ and $0_S$ respectively. If $R$ and $S$ are homomorphic with homomorphism $\phi$ then for all $a \in R$, $\phi (-a) = -\phi(a)$.
• Proof: By Lemma 2 we have that:
(3)
\begin{align} \quad \phi (-a) = \phi(-1 *_1 a) = \phi(-1_R) *_2 \phi (a) = -1_S \phi(a) = -\phi(a) \quad \blacksquare \end{align}