February 10th, 2018, 02:56 AM  #1 
Member Joined: Sep 2011 Posts: 83 Thanks: 1  Ring
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.

February 10th, 2018, 04:16 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 18,574 Thanks: 1485 
If you can't achieve clearer images, I would suggest typing your work.

February 10th, 2018, 08:43 AM  #3 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 119 Thanks: 38 Math Focus: Algebraic Number Theory, Arithmetic Geometry 
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. 
February 10th, 2018, 10:06 AM  #4 
Banned Camp Joined: Apr 2017 From: durban Posts: 22 Thanks: 0 Math Focus: Algebra 
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.

February 10th, 2018, 10:08 AM  #5 
Banned Camp Joined: Apr 2017 From: durban Posts: 22 Thanks: 0 Math Focus: Algebra 
I think you can finish it from here. but looking at the first expression the working is right with Z10

February 10th, 2018, 10:24 AM  #6 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 119 Thanks: 38 Math Focus: Algebraic Number Theory, Arithmetic Geometry  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.
Last edited by cjem; February 10th, 2018 at 10:27 AM. 

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