0

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?

Thanks!

user26857
  • 52,094
chashiey
  • 13

1 Answers1

2

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 $S$:

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

lisyarus
  • 15,517