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 Alexis87 February 10th, 2018 01:56 AM

Ring

3 Attachment(s)
Hi, I have attached the question and the solutions to part a and b of this question. Would like someone to verify if I have done anything wrong especially with part b. Greatly appreciate it! Thanks.

 skipjack February 10th, 2018 03:16 AM

If you can't achieve clearer images, I would suggest typing your work.

 cjem February 10th, 2018 07:43 AM

I'd say this is fine except I'm not sure what you mean by "S is a commutative ring for all $[a], [b] \in Z_{10}$." Surely you just mean it's a commutative ring?

It's worth saying that if a ring $R$ is commutative, then the commutativity of any subring $T$ follows immediately. Indeed, for any $a, b \in T$, note that $a$ and $b$ are in $R$ so they commute. This sort of argument is why we only need to check a few things to show something is a subring - properties like associativity of +, associativity of x, distributivity, follow immediately by the same sort of argument.

 Ola Lawson February 10th, 2018 09:06 AM

The subring test is a theorem that states that for any ring R, a subset of R is a subring if it is closed under multiplication and subtraction, and contains the multiplicative identity of R.

 Ola Lawson February 10th, 2018 09:08 AM

I think you can finish it from here. but looking at the first expression the working is right with Z10

 cjem February 10th, 2018 09:24 AM

Quote:
 Originally Posted by Ola Lawson (Post 588380) The subring test is a theorem that states that for any ring R, a subset of R is a subring if it is closed under multiplication and subtraction, and contains the multiplicative identity of R.
The subset $S \subseteq Z_{10}$ in the question does not contain the multiplicative identity of $Z_{10}$, so it wouldn't actually be a subring under the usual definition (the one you're working with). However, the question still says to prove it's a subring, so they're probably using a different definition.

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