68 729
Assignments Done
100%
Successfully Done
In January 2019

Answer to Question #17647 in Abstract Algebra for Hym@n B@ss

Question #17647
Show that the left regular module R is cohopfian iff every non right-0-divisor in R is a unit. In this case, show that R is also hopfian
Expert's answer
The first statement is clear sinceinjective endomorphisms of RR are given by right multiplications by nonright-0-divisors, and automorphisms of RR are given by rightmultiplications by units. Now suppose non right-0-divisors are units, and
suppose ab = 1. Then xa = 0 =⇒ xab = 0 =⇒ x = 0, soa is not a right-0-divisor. It follows that a ∈ U(R), so we have shown that R is Dedekind-finite. Then RRis hopfian.

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be first!

Leave a comment

Ask Your question

Submit
Privacy policy Terms and Conditions