Question: Question 1 Let R denote a commutative ring, and let I, P R with I P. Show that P is prime in R if and

Question 1

Let R denote a commutative ring, and let I, P R with I P. Show that P is prime in R if and only if P/I is prime in R/I.

Question 2

Let R be an integral domain. Prove that if every ideal of R is a prime ideal then R is a field. (Hint: For any a = 0, a2R is an ideal of R.)

Step by Step Solution

There are 3 Steps involved in it

1 Expert Approved Answer
Step: 1 Unlock blur-text-image
Question Has Been Solved by an Expert!

Get step-by-step solutions from verified subject matter experts

Step: 2 Unlock
Step: 3 Unlock

Students Have Also Explored These Related Mathematics Questions!

Q:

please explain this problem in detail Let R be a commutative ring with unity and p a prime integer. Suppose there exists a nontrivial homomorphism from Z/pZ to R. Show that the map o : R - R defined...

Q:

Every torsion-free divisible abelian group D is a direct sum of copies of the rationals Q. if O n e Z and D, then there exists a unique b D such that nb = a. Denote h by (1/n)a. For m, n Z (n ...

Q:

6. List all the polynomials of degree 2 in Zz[x]. Which of these are equal as functions from Z2 to Zz? 7. Find two distinct cubic polynomials over Z2 that determine the same function from Zz to Z2....

Q:

Exercise 2.133 Let R be a commutative ring. A proper ideal I of R is said to be prime if whenever ab e I for a and b in R, then either a or b must be in I. 1. Show that I is a prime ideal if and only...

Q:

1. (10+5 pts) (a) Let / and J be two left ideals of R and let It J = {xty : re l, ye J}. Prove that I + J is a left ideal of R. (b) Formulate and prove an analogous statement for two-sided ideals. 2....

Q:

6. (i) Show that $N=\{12, 14, 21\}$ generates $\mathbb{Z} $ as a $\mathbb{Z} $-module. [2 Marks] (ii) Decide whether or not a proper subset of $N$ from part (i) [3 Marks] generates $\mathbb{Z}$ as a...

Q:

6. (i) Show that $N=\{12,14,21\}$ generates $\mathbb{Z}$ as a $\mathbb{Z}$-module. [2 marks] (ii) Decide whether or not a proper subset of $N$ from part (i) [3 marks] generates $\mathbb{Z}$ as a...

Q:

6. (i) Show that $N=\{12,14,21\}$ generates $\mathbb{Z}$ as a $\mathbb{Z}$-module. [2 marks] (ii) Decide whether or not a proper subset of $N$ from part (i) [3 marks] generates $\mathbb{Z}$ as a...

Q:

6. (i) Show that $N=\{12,14,21\}$ generates $\mathbb{Z}$ as a $\mathbb{Z}$-module. [2 marks] (ii) Decide whether or not a proper subset of $N$ from part (i) [3 marks] generates $\mathbb{Z} $ as a...

Q:

1. (6 points) Let R be a commutative ring, and let a, b E R. Show that. N ={m+3b | 133 ER} is an ideal of R. 2. (5 points) Let R be a ring and F a eld, and suppose o5 : F > R is a ring homomorphism....

Q:

What is availability and how is it defined?

Q:

Number 12.24 Please help me 12.24. A mass m is suspended by a string of length I from a support S and oscillates as a pendulum in a vertical plane containing the X-axis and making an angle 0 with the...

Q:

How would you do a sensitivity analysis (discussed in Chapter 11) for Fun Toys' net cash balance?

Q:

the shareholders' equity of WBL industries includes the items shown below. The board of directors of WBL declared cash dividends of $14 million, $10 million, and $50 million in its first three years...