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    Ring Theory

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    Rings and Proofs for Commutative Rings

    (See attached file for full problem description with proper symbols) --- 1A) Let R be a commutative ring and let A = {t  R  tp = 0R} where p is a fixed element of R. Prove that if k, m  A and b  R, then both k + m and kb are in A. 1B) Let R be a commutative ring and let b be a fixed ele

    Finitely Generated Z-modules

    Definition: Let R be a commutative ring with identity, let M be an R-module, and let B be a nonempty subset of M. Then the set RB is defined as RB is a submodule. If B is a finite set, say , we write for RB, and say that RB is a finitely generated R-module. In particular, if for some , we say that M is finitely generate

    Ring isomorphism

    (See attached file for full problem description) Z2 means modulo 2 and Z2n means modulo 2n.

    Ring homomorphism

    (See attached file for full problem description) --- 1. Show that if is a ring homomorphism and A is an ideal of R Then need not be an ideal of S. (Compare with property "If A is an ideal and is onto S, then is an ideal).

    Rings, Prime Ideals and Maximal Ideals

    1.Give a example of a commutative ring that has a maximal ideal that is not a prime ideal. 2. Prove that I=<2+2i> is not a prime ideal of Z[i]. How many elements are in Z[i]/I ? What is the characteristic of Z[i]/I ? Please see the attached file for the fully formatted problems.

    Polynomial Rings : Prove Zero - Divisor

    Let A be a commutative ring with identity 1, and let A[x] be the ring of polynomials with coefficients in A. Let f (x) = a0+ a1x + ... + anxn! A[x] . Prove that f (x) is a zero - divisor in A[x] IF AND ONLY IF (iff) 7c ! A, c ! 0 " cf (x) = 0. Please see the attached file for the fully formatted problems.

    Ideals: Prime Ideals and Commutative Ring

    Show that N is contained in P for each prime ideal, P of a commutative ring R. Where N is the set of all nilpotent elements. "a" is nilpotent if a^n=0 for some positive integer n. N itself is an ideal.

    Ring Homomorphism

    See attached pdf file. --- - Find all ring homomorphisms from Z... ---

    Maximal Ideals, Residues and Ring Homomorphisms

    Let . Show that the map the residue of a+ b modulo 2, is a ring homomorphism with . Prove that . Hence, or otherwise, give a maximal ideal of . Consider the ideal (2)+(x) of . Show that (2)+(x) . Hence explain why (x) is not a maximal ideal of . NOTE: All question marks are Z, the integers Please see the attached

    Ring Homomorphisms, Residues and Distinct Maximal Ideals

    Let . Prove that the map given by , where is the residue of a modulo n, is a ring homomorphism. Find the kernel and image of . Prove that if is a ring homomorphism, then given by is also a ring homomorphism. Write down two distinct maximal ideals of . Does have a finite or infinite number of maximal ideals?

    Rings and Fields

    (See attached file for full problem description with proper symbols). For part one....the first is in rational numbers, and second is in integers. ? Verify that is a sub field of and that is a sub ring of . ? Let R be a commutative ring and I an ideal of R. let and be elements of R. prove that is then ?

    Rings, Commutative Rings, Idempotents, Subrings and Isomorphisms

    (1) Given a ring R, an element e is called an idempotent if e^2 = e. (i) Let R1 and R2 be two commutative rings with unity. Consider R = R1 x R2. Find two non-zero idempotents e1,e2 E R such that 1= e1 + e2 and e1e2 = 0. (Be careful: what is 1 in this R? What is 0?) (ii) On the other hand, suppose R is any commutative ring wit

    Ideal rings

    Note: Z is integer numbers C is set containment Here is the problem Let I be an ideal in a ring R. Define [ R : I ] = { r in R such that xr in R for all x in R } 1) Show that [ R : I ] is an ideal of R that contains I 2) If R is assumed to have a unity, what can you say about [ R : I ] ? 3) Find [ 2Z :

    Modules, Submodules, Commutative Rings with Unity

    1. Let R and S be commutative rings with unity, and let &#966;: R S be a ring homomorphism. If M is an S-module, prove that M is also an R-module if we define rm = &#966;(r)m for all r E R and m E M. 2. If M1 and M2 are submodules of an R-module M such that M = M1(+)M2, prove that M1 = M/M2 and M2 = M/M1.

    Rings and Modules : Quasi-Regular, Module Homomorphisms

    1. Let R be a ring. Prove that if x, y E R such that xy is right quasi-regular then yx is also right quasi-regular. 3. Let M and N be left R-modules. Let f : M N and g : N M be left R-module homomorphisms such that fg(y) = y for all y N. Show that M = ker(f) im(g). Please see attached.

    Ring Identities Equality Properties

    For any x,y E R (i.e. Ring R) the following equalities hold. a) 0.x=0 b) a(-b)=(-a)b=-(ab) Prove either a or b. State any properties used in your proof. Please see attached for full question.

    Ideals of a Ring : Containment and Subgroups

    Let J and I be ideals of the ring R, with J C I C R. Show that J is an ideal in the ring I (Recall that any ideal of a ring is also a subring; so I is a ring in its own right) Note that: To show J an ideal of I, we must show that 1) it is an additive subgroup of R and 2) it is satisfied bJ C J and Jb C J

    Rings of Unity, Monoid, Momomorphism and Invertible Elements

    Let R commutative ring with unity, and S a sub monoid of the multiplicative monoid of R. In RxS define (a,b) ~ (b,t) if &#1069;u &#1108; S &#1101; u(at-bs)= 0. Show that ~ is an equivalence relation in RxS. Denote the equivalence class of (a,s) as a/s and the quotient set consisting of these classes as RS-¹. Show that RS-¹ be

    Rings with Unity, Isomorphism, Bijectiveness and Invertibility

    Let R be a ring with unity e, R' a set, H a bijective map from R' into R. show that R' becomes a ring with unity if one defines: a'+ b' = Hˉ¹ (H(a')+ H(b')) a' b' = Hˉ¹(H(a') H(b')) 0'= Hˉ¹(0) e'= Hˉ¹(e) and that is an isomorphism of R' with R. Use this to prove that if u is an invertible element of a ring

    Ring Homomorphisms

    Let phi be a homomorphism of a ring R with unity onto a nonzero ring R'. Let u be a unit in R. Show that phi (u) is a unit in R'.