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Determine whether or not a mapping is a ring homomorphism


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.3 Exercise 7.3.6

Decide which of the following are ring homomorphisms from $M_2(\mathbb{Z}) to \mathbb{Z}$.

(1) $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \mapsto a$ (Projection onto the $(1,1)$ entry.)
(2) $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \mapsto a+d$ (Trace)
(3) $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \mapsto ad-bc$ (Determinant)


Solution:

(1) Let $A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$. Evidently, $\varphi(AA) = 2$, while $\varphi(A) \varphi(A) = 1$; thus this mapping is not a ring homomorphism.

(2) Again let $A = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$. Evidently, $\varphi(AA) = 3$, while $\varphi(A)\varphi(A) = 1$; thus this mapping is not a ring homomorphism.

(3) Let $A = \begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix}$. Evidently, $\varphi(A+B) = 1$ while $\varphi(A) + \varphi(B) = 0$. Thus this mapping is not a ring homomorphism.


Linearity

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