Let R be a ring and let U and V be towsided ideals of R whic

Let R be a ring and let U and V be (tow-sided) ideals of R. which of the following must also be ideals of R? Explain

1. U + V = { u+v: uU and vV }

2. U V = { uv : uU and vV }

3. U V

A) 2 only   B) 3 only C) 1 and 2 only D) 1 and 3 only E) 1,2, and 3

Solution

1. The sum of two ideals of a ring R is again an ideal

2. If U and V are ideals of ring R, then UV is again an ideal of ring R

3. intersection of two ideals is again an ideal

therefore, the correct option is (E)