

Let R be Commutative ring with 1 and let N and M be two R-modules Prove...
11. (a) Let F be a field. Prove FixF Rİr (b) Let R be a commutative ring with identity. Prove that one can have R.
11. (a) Let F be a field. Prove FixF Rİr (b) Let R be a commutative ring with identity. Prove that one can have R.
Let R be a commutative ring with 1. Prove that Ris a field if and only if the only ideals in Rare (0) and R.
2. Let R be a commutative ring with unity 1, and let a be a unit in R Let / be an ideal in R that contains the element a. Prove that / cannot be a proper ideal of R. 3. Let R be a commutative ring with unity 1 of order 30, and let be a prime ideal of R. Prove that is a maximal ideal of R
thanks
Let I be a proper ideal of a commutative ring R with 1. Prove that I is a maximal 3. (10 ideal of R if and only if for every a e R\I, I+(a) : {i+ ar i e I,rE R} = R.
Let I be a proper ideal of a commutative ring R with 1. Prove that I is a maximal 3. (10 ideal of R if and only if for every a e R\I, I+(a) : {i+...
Abstract Algebra
(6) Let R be a commutative ring. For elements r, s є R, prove the Binomial Theorem in R: Here if n Z, r є R, we interpret nr to be the element in R which is a sum f n many r's.
(6) Let R be a commutative ring. For elements r, s є R, prove the Binomial Theorem in R: Here if n Z, r є R, we interpret nr to be the element in R...
Part 1
Part 2
7.1.2. Let R be a commutative ring and a, b E R, and define The goal of this problem is to prove that (a, b) is an ideal of R (a) Explain how you know that 0 E (a, b b) What do two random elements of (a, b) look like? Explain why their sum must be in (c) For s E R and z E (a,b), explain why sz E (a, b). 7.2.1. In the...
Additional Problems: Let R be a commutative unital ring, and let M be an R-module. (1) The set (reRIrm 0 for all m e M) is called the annihilator of M. Show that it is an ideal of R. (2) Show that the intersection of any nonempty collection of submodules of M is a sub- module.
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].
(5) Suppose R is a commutative ring with unity, and r e R. Let A(r) {s E R : rs-0). Prove that A(r) is an ideal of R.