Suppose R is a commutative ring with 1_R. Let P be a prime ideal of R, I be the principal ideal generated by a element of R\P. If P subset of I, then which one is true? which one is wrong? Explain each part. (i) P = aP (ii) aP subset P (iii) P + I = R (iv) P is a maximal ideal of R.

Suppose R is a commutative ring with 1_R. Let P be a prime ideal of R, I be the principal ideal generated by a element of R\P. If P subset of I, then which one is true? which one is wrong? Explain each part. (i) P = aP (ii) aP subset P (iii) P + I = R (iv) P is a maximal ideal of R.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Suppose R is a commutative ring with 1_R. Let P be a prime ideal of R, I be the principal ideal
generated by a element of R\P. If P subset of I, then which one is true? which one is wrong? Explain each part.
(i) P = aP
(ii) aP subset P
(iii) P + I = R
(iv) P is a maximal ideal of R.

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