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Let Prove or disprove that the mapping is a ring

Chapter 15, Problem 16E

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QUESTION:

Let \(R=\left\{\left[\begin{array}{ll}a & b \\ 0 & c\end{array}\right] \mid a, b, c \in Z\right\}\). Prove or disprove that the mapping \(\left[\begin{array}{ll}a & b \\ 0 & c\end{array}\right] \rightarrow a\) is a ring homomorphism.

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QUESTION:

Let \(R=\left\{\left[\begin{array}{ll}a & b \\ 0 & c\end{array}\right] \mid a, b, c \in Z\right\}\). Prove or disprove that the mapping \(\left[\begin{array}{ll}a & b \\ 0 & c\end{array}\right] \rightarrow a\) is a ring homomorphism.

ANSWER:

Step 1 of 3

We denote given mapping with . It is obviously additive, so we only need to verify that  preserves multiplication.

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