Basic properties of left and right units and left and right zero divisors

Let R be a ring with 1 \neq 0. A nonzero element a \in R is called a left zero divisor in R if there is a nonzero element x \in R such that ax = 0. Symmetrically, b \neq 0 is called a right zero divisor in R if there is a nonzero element y \in R such that yb = 0. (So a zero divisor is an element which is either a left or a right zero divisor, or both.) An element u \in R is said to have a left inverse in R if there is some s \in R such that su = 1. Symmetrically, v has a right inverse if there exists t \in R such that vt = 1.

  1. Prove that u \in R is a unit if and only if it has both a right and a left inverse.
  2. Prove that if u has a right inverse then u is not a right zero divisor.
  3. Prove that if u has more than one right inverse then u is a left zero divisor.
  4. Prove that if R is a finite ring then every element that has a right inverse is a unit (i.e. has a two sided inverse).

  1. First, suppose u \in R is a unit. Then there exists v \in R such that uv = vu = 1; thus v is both a left and a right inverse for u. Suppose now that uv = 1 and wu = 1. (That is, that v is a right inverse of u and w is a left inverse.) Now v = 1 \cdot v = w \cdot u \cdot v = w \cdot 1 = w, so that v = w. Thus u has a two sided inverse, and thus is a unit.
  2. Suppose u \neq 0 has a right inverse and is a right zero divisor; that is, uv = 1 for some v \in R and zu = 0 for some z \in R, where z \neq 0. Then 0 = zu implies 0 = zuv, so that z = 0, a contradiction. So if an element has a right inverse, it is not a right zero divisor.
  3. Suppose u \in R has distinct right inverses v and w– that is, uv = uw = 1, and v \neq w. Now u \neq 0 and v-w \neq 0, but u(v-w) = uv-uw = 1-1 = 0, so that u is a left zero divisor.
  4. Let R be a finite ring. Let u \in R have a right inverse; say uv = 1. Now define \varphi : R \rightarrow R by \varphi(x) = xu. Suppose \varphi(x) = \varphi(y); then xu = yu, so that xuv = yuv, and we have x = y. Thus \varphi is injective. Since R is finite, in fact \varphi is surjective. Thus \varphi(x) = 1 for some x \in R, and we have xu = 1. Since u has a left and right inverse, by part 1 above, u is a unit in R.
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