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I am trying to do the following question, from Pinter's "A Book of Abstract Algebra":

Prove that A satisfies all the axioms to be a commutative ring with unity. Indicate the zero element, the unity, and the negative of an arbitrary a.

$1)$ A is the set of the integers, with the following “addition” $\oplus$ and “multiplication $\times$” :

$$a \oplus b = a + b−1$$

$$a \times b = ab − (a + b) + 2$$

I have never come across rings before and so I have no clue where to start.

I'd like some tips as there aren't any "how to show something is a ring" examples in the textbook. Thank you for any help!

Bill Dubuque
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Adam Mac.
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    Show that the axioms (associative, commutative "addition", distributive, etc.) are satisfied. For example (associative), is $(a ⊕ b) ⊕ c=a ⊕ (b ⊕ c)"? Can you figure out what the additive and multiplicative identity elements are? – J. W. Tanner Jul 12 '21 at 17:06
  • Yes, but how do I do that? Sorry if that is a stupid question! – Adam Mac. Jul 12 '21 at 17:08
  • @AdamMac. Have you studied groups or vector spaces? This is no different than showing that a set is one of those. – John Douma Jul 12 '21 at 17:11
  • I think, following J.W. Tanner's comment, that it is easier to proceed if you first figure out what the additive and multiplicative identities are. An additive identity is one of the integers, call it $z$, such that $z \oplus x = x \oplus z = x$ for any value of $x$ whatever. Well, if we carry out the definition of $\oplus$, we get $z \oplus x = z+x-1$. If that is equal to $x$, then $z+x-1 = x$, and then $z =$ what? Similarly, there must exist an integer $u$ such that $u \times x = x \times u = x$ for any $x$. What must $u$ be equal to? – Brian Tung Jul 12 '21 at 17:21
  • As in the linked dupes, transport structure via $,h(x) = x-1\ \ $ – Bill Dubuque Jul 12 '21 at 18:07

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Basically you "simply" have to check if the operations satisfy the axioms for commutative rings with unit, for instance if the following equality holds: $$ x \oplus (y \oplus z) = (x \oplus y) \oplus z$$.

To verify if this equation holds you just need to use the definition of $(\oplus)$, namely the equation $$a \oplus b = a+b-1\ ,$$ to expand the expressions $x \oplus(y \oplus z)$ and $(x \oplus y) \oplus z$ to the corresponing numeric expression and check using ordinary integer arithmetic if these derived expressions are actually equal or not.

Of course you have to check the other axioms as well.

I hope this helps.

Giorgio Mossa
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