# Prove properties of a ring with additive identity 0.

Let R be a ring with additive identity 0. Prove the following:

(a) For all a in R, a(0) = 0.

(b) a(-b)=-(ab).

NOTE: see attached word document for clearer notations.

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#### Solution Preview

First 0+0 =0, in any ring;

next, we observe that

a(0 + 0) = a0 = 0

to see this, use the distributive property on the left side to get

a0 + a0 = ...

#### Solution Summary

Properties of a ring with additive identity 0 are proven. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Abstract Algebra (4 year College)

I had to prove 4 theorems two of them dealt with Abelian elements, automorphism of R2 under compontentwise addition. I want to keep my original work in tack as possble BUT I would like the follwing corrections made based on the following comments.

THESE ARE THE ISSUES THAT NEED TO BE ADDRESSED

The paper needs to provide a more comprehensive response to the problem. (The first page lists the directions.)

The comments I was given to improve the paper and that needs to be addressed is as follows:

For each proof, my answer needs to clearly state relevant definitions (Abelian Group, Automorphism, Ring, Integral Domain). Such definitions should be incorporated within each proof making all appropriate connections.

The paper should provide well-written justifications for relevant steps using complete and comprehensive sentences.

More details need to be shown with regard to algebraic manipulations via the proper use of parentheses.

The paper should elaborate further on the difference between being ONE-TO-ONE versus ONTO. It is important for the paper to convey to the reader a complete understanding of this difference.

The paper provides reasoning which is appropriate for elements belonging to the set of all real numbers, but recall that such reasoning should be focused on the set of elements belonging to Zn. This must be redone to show what is stated.

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