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The diagonal subset of the Cartesian square of a ring is a subring


Solution to Abstract Algebra by Dummit & Foote 3rd edition Chapter 7.1 Exercise 7.1.18

Let $R$ be a ring. Prove that $\Delta(R) = \{(r,r) \ |\ r \in R \}$ is a subring of $R \times R$.


Solution: In this previous exercise, we saw that $\Delta(R)$ is an additive subgroup of $R \times R$; thus it suffices to show closure under multiplication. To that end, if $r,s \in R$, then $$(r,r)(s,s) = (rs,rs) \in \Delta(R).$$


Linearity

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