It is a review problem for abstract algebra, and I'm having trouble figuring it out.

Ring $R=\mathbb Z_4\times\mathbb Z_6$, which is a commutative ring with unity $(1,1)$. Find the units in this ring.

I know how to find units when $R=Z_4$ and $R=Z_6$ separately, but I don't know when it comes to $R=Z_4$ x $Z_6$. Should I list all the elements in $R$ first?



Recall that units are just invertible elements, and in ring product multiplication is done componentwise. So, the units of $R\times S$ are precisly pairs of units of $R$ and $S$:

$$r\cdot r'=\mathbb{1}_R$$ $$s\cdot s'=\mathbb{1}_S$$ $$(r,s)\cdot (r',s')=(r\cdot r', s \cdot s') = (\mathbb{1}_R, \mathbb{1}_S) = \mathbb{1}_{R\times S}$$


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.