Let u be a unit in R 1 Prove that u notequalto 0 You may use

Let u be a unit in R. 1. Prove that u notequalto 0. (You may use the fact that r0 = 0r m 0 for all r R.) 2. Prove that u is not a Zero divisor in R. That is. if it u is a unit and r R is such that ur = ru = 0, then r = 0. 3. Prove that if v,v\' R and uv = vu = 1 = v\'u = uv\'. the element v = v\' is denoted by u^-1 R.

Solution

(1) Let an element u of R is a unit.

if there is an element bR such that ub=bu=1

the element b is called multiplicative inverse for u so we can write b = u^-1

0 can never be a unit.

by the definition 0 1 and 0.b = 01 for all b in R.

therefore u 0.

(2) Let U be the unit in R with multiplicative inverse u^-1

suppose ur = 0 for some rR.

then u^-1(ur)=u^-10 (well defined)

(u^-1u) r = u^-10 (associative under multiplication)

1r = u^-10 (multiplicative inverse)

r= 0 (multiplicative identity)

therefore if ur=0 for some rR then r = 0

Similarly we can show that if ru = 0 for some rR then r = 0.

Hence u is not a zero divisor.

(3) Since u is a unit in R

if there exist v in R such that uv = vu = 1

if there exist v¹in R such that uv¹=u¹v = 1

therefore v and v¹ are multiplicative inverses of u.

Hence v = v¹=u^-1.