{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# ahw7 - Math 5125 Monday October 17 Seventh Homework...

This preview shows pages 1–2. Sign up to view the full content.

Math 5125 Monday, October 17 Seventh Homework Solutions 1. Note that n Z m Z because m n . By the ideal correspondence theorem we now obtain Z / n Z m Z / n Z = Z / m Z . However the ideals m Z / n Z and m ( Z / n Z ) of Z / n Z are equal, because they both con- sist of the cosets mr + n Z for some r Z , and the result follows. 2. Section 7.3, Exercise 35 on page 250. Let I , J , K be ideals of R . (a) Prove that I ( J + K ) = IJ + IK and ( I + J ) K = IK + JK . (b) Prove that if J I , then I ( J + K ) = J +( I K ) . (a) Obviously IJ , IK I ( J + K ) and since I ( J + K ) is closed under addition, we see that IJ + IK I ( J + K ) . On the other hand the general element of I ( J + K ) is of the form r i r ( j r + k r ) , where i r I , j r J and k r K , and this can be written in the form r i r j r + r i r k r , which shows that I ( J + K ) IJ + IK and it follows that I ( J + K ) = IJ + IK . The proof that ( I + J ) K = IK + JK is exactly similar. (b) Obviously J I (given), J J + K , I K I and I K J + K . Since I ( J + K ) is closed under addition, we deduce that J +( I K ) I ( J + K ) . Now suppose

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern