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i am wondering if the unity elements of a ring form a ring ? In other words do they form an abelian group under addition ? I have tryied but i have not reached to a conclusive answer. Thanks for any comment.

user249018
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  • How do you define "unity". With what I'd expect to be the normal definition (the identity) the answer is obviously "no" in most cases - the only cases is rings where $x+x=0$ in general. – skyking Oct 30 '17 at 10:20

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Is $0$ a part of your new "ring of unity elements"? Because it has to be if you want to call what you have a ring.

If you choose to include $0$, try with a few examples. How does this work for the archetypal ring, $\Bbb Z$, for instance? This should give you the answer you're looking for.

Also, there is a lot more to a subset being a subring than just forming an abelian group under addition. Otherwise there would be little difference between the theory of rings and the theory of abelian groups.

Arthur
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  • Thanks. 0 has no inverse under multiplication and thus is not a uniy element. You are saying the ring of all rings is $ \Bbb Z$, which i dont understand. Is $ \Bbb Z$ the ring of all rings ? – user249018 Oct 30 '17 at 09:51
  • @user249018 That was an incredibly unfortunate choice of words. I have rephrased. I meant to convey that $\Bbb Z$ is the Most Important Ring, as in it fulfills some important universal properties, it is well-known, most people have good intuition about how it works, and it serves as a counterexample to many "niceness" properties that one could want rings to have. Whenever you feel like you have an idea for something that should be true for all (commutative) rings, $\Bbb Z$ is always the first one you should check. – Arthur Oct 30 '17 at 10:33
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In a ring the element $0$ is never invertible, thus the set of units is never a ring.

Besides, the sum of units is not necessarily a unit: e.g. $1+1=2$ is not invertible in $\Bbb Z$.

What is true is that the set of units forms a group (under multiplication: think of $\Bbb R^\times$ or $\{\pm1\}$).

Andrea Mori
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