Question: 6. Let R be a commutative ring with unity such that r2 = r for all r E R. Let I be a prime idea]

6. Let R be a commutative ring with unity such that r2 = r for all r E R. Let I be a prime idea] of R. Prove that R/I consists of exactly two cooets: I and 1 +

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!

Let R be a commutative ring with unity. If I is a prime ideal of R , prove that I[x]is a prime ideal of R[x] .

Let R be a commutative ring with unity of characteristic p a prime number. Prove that the function Frob(x)=xp is a ring homomorphism. Prove that if R is a finite field then Frob is an isomorphism....

I need help with this problem in abstract algebra please, and please, please make it legible I often can't read the handwriting. 2. Let R be a commutative ring with unity. If / is a prime ideal of R,...

Let R be a commutative ring with unity u, and let I be an ideal of R. (a) If u 1, prove that I = R. (b) If I contains a unit of R, prove that I = R.

Let R be a commutative ring with unity Let X be a subset of R. Prove that there exists a smallest ideal in R which contains X. (An ideal is said to be smallest ideal satisfying a given condition c,...

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....

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....

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...

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...

Let R be a commutative ring with identity of prime characteristic p. If a,b R, then (a b) p n = a pn b pn for all integers n 0 [see Theorem 1.6 and Exercise 10; note that b = -b if p = 2]....

Protein synthesis takes place much more slowly in eukaryotes than in prokaryotes. Suggest a reason why this is so.

Carton Corporation produces a line of beverage cartons. The production process has been automated, so the product can now be produced in one operation rather than in the three operations that were...

Complete the sentences by choosing the correct response from each of the drop - down lists. The earnings per share ( EPS ) ratio shows how much an investor can benefit from paying taxesholding each...

A company issues 1,050 shares of its common stock for $33,600 cash. Prepare journal entries to record this event under each of the following separate situations.