Solved: For the following statement S and proposed proof, either (1) S is true and the

Chapter 14, Problem 14.25

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For the following statement S and proposed proof, either (1) S is true and the proof iscorrect, (2) S is true and the proof is incorrect, or (3) S is false and the proof is incorrect.Explain which of these occurs.S: Let R be a ring with unity containing at least two elements and letR = {a R : a r is a unit for each r R}.Then R is a subring of R.Proof. Let a, b R. First, consider a b and r R. Then (a b) r = a (b + r).Since a R and b + r R, it follows that (a b) r is a unit and so a b R. Next,consider ab and r R. Then ab r = a (a ab + r). Since a R and a ab + r R,it follows that ab r is a unit. Thus ab R. By the Subring Test, R is a subring of R.