1 answer

Suppose R is a ring with a least two elements. Suppose that for any r not equal...continues

Question:

Suppose R is a ring with a least two elements. Suppose that for any r not equal to 0r, there is a unique s in R such that rsr = r. Prove each of the following:(a) R has no zero divisors(b) srs=s(c) R has a unity element

Answers

(a) Suppose r,s,t, are elements in R such that

rsr=r, and rx=0 or xr=0. Then

r=rsr=rsr-rxr=r(s-t)r.

s is supposed to be unique in R to satisfy rsr=r, so we see that

s-t=s, or t=0. As we can show this for all r in R, it follows that there are no zero divisors in R.

(b) Suppose r,s in R such thatrsr=r

Then

rs=(rsr)s=r(srs)

sors−r(srs)=0

sor(s−srs)=0

Because r cannot be a zero divisor,

s-srs=0, and so srs=s.

(c) Suppose t an element in R. Then

rt=rsrt

so rt-rsrt=0

so r(t-srt)=0

Because r is not a zero divisor,

t-srt=0 so srt=t, so sr is a left unity in R.

Now, look at ts=tsrs

so ts-tsrs=0

so t(s-srs)=0

Because r is not a zero divisor,

s-srs=0 so srs=s

Thus sr is both a left and right unity in R, which follows that sr is the unity element in R.

.

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