Answer to Question #148645 in Abstract Algebra for Mayank Jain

Question

There is an injective ring homomorphism from M2 [Z] to M2[Z].Is it true?

Answer 1

Define a map $\varphi : M_2[\mathbb Z]\to M_2[\mathbb Z],\ \ \varphi(A)=A.$

Then $\varphi(A+B)=A+B=\varphi(A)+\varphi(B)$ and $\varphi(A\cdot B)=A\cdot B=\varphi(A)\cdot\varphi(B)$. Thus $\varphi$ is a ring homomorhism. If $A\ne B$, then $\varphi(A)=A\ne B=\varphi(B)$, and consequently, $\varphi$ is injective.

Answer: true