Thread: Rings with Unity

I know that unity for a ring is just the multiplicative inverse for a ring, and that not every ring has a multiplicative inverse, so therefore not every ring has unity. I need to prove why a ring has at most one unity. I'm just not sure how to construct a well written proof for this. Any help would be greatly appreciated.

Originally Posted by tfhawk

I know that unity for a ring is just the multiplicative inverse for a ring, and that not every ring has a multiplicative inverse, so therefore not every ring has unity. I need to prove why a ring has at most one unity. I'm just not sure how to construct a well written proof for this. Any help would be greatly appreciated.

I'm not quite what you mean by multiplicative inverse for a ring?

Unity is the multiplicative Identity

PlanetMath: unity

The idea of the proof is

what if there are to elements of the Ring that are the multiplicative Identity call them

but then

becuase is an identity, and

becuase

This implies that



So the identity is unique.

I think you are confusing the term "unit" and "unity" which is understandable. A unit is an element in a ring that has an inverse in the ring (multiplicative).
unity is the identity element 1, such that 1r=r for all r in your ring. It is indeed uniques as pointed out.

It should also probably be noted that rings can certainly have unity without every element having an inverse. For instance is a ring with unity 1, but only 1 and -1 have inverses.

This is also a typical example to show that not every element in a ring is either a unit (invertible) or a zero divisor. This is an integral domain and has no zero divisors and only 2 units.