Answer to Question #165451 in Abstract Algebra for Samia

Question

prove that the element (u,v) in direct product r×s of two rings r and s is unit if and only if u is a unit in r and v is unit in s

Answer 1

Let's first show, that $(u,v)$ is an identity element. Indeed, by definition of ring structure on a product $R\times S$ we have:

$(u,v)\cdot(r,s)=(u\cdot r,v\cdot s)=(r,s)$

$(r,s)\cdot(u,v)=(r\cdot u,s\cdot v)=(r,s)$

Now as identity is unique, the identity element on $R\times S$ should forcefully be $(u,v)$.