### Ring Theory : Abelian Groups, Nilradicals and Augmentation Ideals

Let p be a prime and let G be an abelian group of order a power of p. Prove that the nilradical of the group ring G is the augmentation ideal.

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Let p be a prime and let G be an abelian group of order a power of p. Prove that the nilradical of the group ring G is the augmentation ideal.

1.a) Let R be a ring with 1 and let S=M2(R). If I is an field of S, show that there is an ideal J of R such that I consists of all 2X2 matrices over J. 1 b) Use the result of 1 a) to prove the following question. Let R be the ring of 2X2 matrices over reals; suppose that I is an ideal of R. Show that I =(0) or I=R.

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Prove that the units in a commutative ring with a unit element form an abelian group.

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Let I be a non-empty index set with a partial order <=, and A_i be a group for all i in I. Suppose that for every pair of indices i,j there is a map phi_ij:A_j ->A_i such that phi_jiphi_kj= phi_ki wheneveri<=j<=k, and phi_ii=1 for all i in I. Let P be the subset of elements(a_i) with i from I in the direct product D of A_i such

Let I be a non-empty index set with a partial order<=. Assume that I is a directed set, that is, that for any pair i,j in I there is a,k in J such that i<=k and j<=k. Suppose that for every pair of indices i,j with i<=j ther is a map p_ij: A_i->A_j such that p_jkp_ij=p_ik whenever i<=j<=k and p_ii=1 for all i in I. Let B be the

Let m and n be positive integers with n dividing m. Prove that the natural surjective ring projection Z/mZ ->Z/nZ is also surjective on the units:(Z/mZ)^x ->(Z/nZ)^x

If F is a field, prove that the field of fractions of F[[x]] is the ring F((x)) of formal Laurent series. Show that the field of fractions of the power series ring Z[[x]] is properly contained in the field of Laurent series Q((x)). Ps. Here F[[x]] is the ring of formal power series in the indeterminate x with coefficients in

Let phi:R->S be a homomorphism of commutative rings a) Prove that if P is a prime ideal of S then either phi^-1(P)=R or phi^-1(P) is a prime ideal of R. Apply this to the special case when R is a subring of S and phi is the inclusion homomorphism to deduce that if P is a prime ideal of S then PR is either R or prime ideal in

Prove that a necessary and sufficient condition that the element 'a' in the Euclidean ring is a unit is that d(a) = d(1). Or, Prove that the element 'a' in the Euclidean ring is a unit if and only if d(a) = d(1).

Modern Algebra Ring Theory (IX) The Field of Quotients of an Integral Domain Prove that the mapping φ:D→F defined by φ(a) =

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(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

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

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.

If n Є R and R is a commutative ring we indicate by Mn(R) the ring of allnxn entries wrt the usual operations on matrices. If n>1 this ring is commutative even if R is. Let S={(aij)ЄMn(R)|i≠j=>aij=0} Let k be an integer 1≤k≤n. Show that a) S is a commutative subring of Mn(R) b) The function f: S

Let R be the set of all continuous functions from the set of real numbers into itself. Then R is a commutative ring with the following operations: (f+g)(x)=f(x) + g(x) and (fg)(x)=f(x)g(x) for all x. Now let I be the set of all functions f(x) an element of R such that f(1)=0. Show that I is a maximal ideal of R. Maximal

See attached pdf file.

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

Prove that a module over a polynomial ring C[t] is a finite dimensional vector space with a linear operator that plays the role of multiplication by t.

(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

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 :

Modern Algebra Division Ring

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 Эu є S э 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