rev2 - Math 5125 Friday October 28 Second Test Review...

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

View Full Document Right Arrow Icon
Math 5125 Friday, October 28 Second Test Review Sample Second Test no. 3. Answer All Problems. Please Give Explanations For Your Answers. 1. Let R be a commutative ring which has nonprincipal ideals. Prove that R has a max- imal nonprincipal ideal (that is, an ideal I of R such that I is not principal, but if I J ± R , then J is principal). (16 points) 2. Prove that 2, 3 ± 13 are irreducible in Z [ 13 ] . Also prove that 2 is not prime in Z [ 13 ] . (17 points) 3. Let R be a UFD and let P be a nonzero principal ideal of R . (a) Prove that there are only finitely many principal ideals of R containing P . (b) Is it true that there are only finitely many principal ideals of R / P ? (Prove or give counter example.) (17 points) Solution to Problem 2 on Sample Test no. 1 First note that S - 1 M ± S - 1 R . Obviously 0 S - 1 M . If m / s , n / t S - 1 M , then m / s + n / t = ( mt + ns ) / ( st ) S - 1 M . Also if r / u S - 1 R , then ( m / s )( r / u ) = ( mr ) / ( su ) S - 1 M . If S M 6 = /0, choose s S M . Then s / s = 1 S - 1 M , which shows that S - 1 M = S - 1 R and hence S - 1 M is not a maximal ideal. Now suppose S M = /0. If S - 1 M = S - 1 R , then 1 / 1 = m / s for some m M and s S . This tells us that ( m - s ) t = 0 for some t S and we deduce that st M which contradicts S M = /0. Therefore S - 1 M 6 = S - 1 R . Finally suppose S - 1 M J ± S - 1 R and J 6 = S - 1 R . Let θ : R S - 1 R denote the natural homomorphism r 7→ r / 1. Then M θ - 1 J ± R and θ - 1 J 6 = R . Therefore θ - 1 J = M by maximality of M . If x J , then x = y / s for some s S and y R , and y / 1 J θ ( R ) . Therefore y θ - 1 J and we deduce that y M . Then x = y / s S - 1 M , consequently J = S - 1 M and we conclude that
Background image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 3

rev2 - Math 5125 Friday October 28 Second Test Review...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online