Examples of Rings

DgrayMan · **Joined:** Sat, 21 Jan 2012 03:59:22 UTC **Posts:** 182

So here's the problem: In each of the following, a set A w/ operations addition and multiplication is given. Prove that A satisfies all the axioms to be commutative ring w/unity. Indicate the zero element, the unity and the negative of an arbitrary a.
1.) A is the zet of Z of integers with the following "addition" + and "multiplication" x:

a+b=a+b+1 ; axb=ab-(a+b)+2.

So i worked it out but my teacher said i need to show the distributive property?

do i have to multiply by an x, like x(a+b+1) and x((a+b)+2)?

Shadow · **Posted:** Fri, 4 May 2012 16:29:14 UTC

DgrayMan wrote:

So here's the problem: In each of the following, a set A w/ operations addition and multiplication is given. Prove that A satisfies all the axioms to be commutative ring w/unity. Indicate the zero element, the unity and the negative of an arbitrary a.
1.) A is the zet of Z of integers with the following "addition" + and "multiplication" x:

a+b=a+b+1 ; axb=ab-(a+b)+2.

So i worked it out but my teacher said i need to show the distributive property?

do i have to multiply by an x, like x(a+b+1) and x((a+b)+2)?

I'll let $\oplus$ denote your version of addition and $\otimes$ denote your version of multiplication so that $a\oplus b = a+b+1$ and $a\otimes b = ab-a-b+2$

Then you need to show that $a\otimes (b\oplus c)=a\otimes b \oplus a\otimes c$ and if you have shown commutativity of $\otimes$ that's enough, if not, you also need to show $(a\oplus b)\otimes c = a\otimes c \oplus b\otimes c$

S.O.S. Mathematics CyberBoard

Examples of Rings

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