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# Ring Theory

### Local ring proof

A local ring is commutative with identity which has a unique maximal ideal. Prove that R is local if and only if the non-units of R form an ideal. I need a detailed, rigorous proof of this with explanation of the steps so I can learn. Thank you.

### Group and Ring Homomorphisms

Please attached for the question to help me understand Group and Ring Homomorphisms. Is this function a Homomorphism? a) Does &#61538;1:&#61530;22 &#61614;:&#61530;22 where &#61538;1(a1,a2) = (a1,0) define a group homomorphism? a ring homomorphism? Prove your answers. b) Does &#61538;1:&#61530;22 &#61614;:&#61530;22

### Ring and ideal proofs

** Please see the attached file for the complete problem description ** Let R be a ring....Prove that radical I is an ideal.....Prove that there is a prime ideal P containing I....

### Verifying Laws Using Ring Fundamentals

For the ring M2(Z) described in Example 3.6. verify the associative law of multiplication and the distributive laws. What is the zero of this ring? Verify that the following is the identity of M2(Z). Give examples to show that M2(Z) is a noncommutative ring. 1 0 0 1 Please see attached files for full problem descrip

### Finding the Additive Inverse and Verifying the Subset of a Ring

What is the additive inverse of each element in the ring R described in Example 3.4? Verify that the subset {u,v} of the ring R in Example 3.4 is a subring. Show that after a change of notation this is the same ring as described in Example 3.5. For the ring R described in Example 3.4, use the tables to verify each of the f

### Rings: Addition and Multiplication

Which of the following are rings with respect to the usual definition of addition and multiplication? (see attached)

### Commutative Ring Proof

Let R be a commutative ring with 1. Prove that if (a,b)=1 and a divides bc, then a divides c. More generally, show that if a divides bc with nonzero a,b then a/(a,b) divides c. (Here (a,b) denotes the g.c.d. of a and b).

### Field and ring proofs

1. Define F sub four to be the set of all 2x2 matrices. F(sub 4)= [ a b ] ; a,b elements of F sub 2 b a+b i) Prove that F sub four is a commutative ring whose operations are matrix addition and matrix multiplication ii) prove that F sub four is a field having exactly four elements iii) show that I sub f

### Rings

1. a. show that every subfield of complex numbers contains rational numbers b. show that the prime field of real numbers is rational numbers c. show that the prime field of complex numbers is rational numbers 2. a. Let R be a domain. Prove that the polynomial f(x) is a unit in R[x] if and only if f(x) is a nonzero cons

### Ring Proof: Idemopotents and Nilpotents

Let R = C([0, 1]) be the ring of continuous real-valued functions on the interval [0, 1], with the usual definitions of sum and product of functions from calculus. Show that f in R is a zero divisor if and only if f is not identically zero and { x | f(x) = 0 } contains an open interval. What are the idempotents of this ring? W

### Algebraic Number Theory - Definitions

Briefly define a group. Briefly define a ring. Briefly define a field.

### Algebraic Number Theory Element

Recall that an element a is called algebraic over a ring (or field) R if a is the root of some polynomial in R[x]. Is the element a = 2/5 + i/3, (R= the integers) algebraic over the ring (or field) R? If so, give a polynomial in R[x] which has a as a root.

### Logic and validity of arguments

Hi Shrikant, I wonder if your can help me with this: 1) Determine whether the argument is valid or invalid. A tree as green leaves or the tree does not produce oxygen. This tree has green leaves. ------------------------------------------------------------- Therefore:. This tree does not produce oxyge

### Ring Isomorphism and the Evaluation Homomorphism

Please provide details of this proof.

### Polynomial Rings and Composition Series

Let R = Q[X,Y] be the polynomial ring in two variables. Show that the R-module R/(X^2 -Y) has no composition series.

### Polynomial Rings Appreciated

Any help is greatly appreciated, I assure you. Thank you!

### Unique Ring Homomorphisms

Let F be a field. Let . Prove that there exists a unique ring homomorphism such that . Please see the attached file for the fully formatted problems.

### Polynomial Rings, Module Endomorphisms and Field Isomorphisms

Please see the attached file for the fully formatted problems.

### Local Rings and Maximal Ideals

A commutative ring R is called a local ring if it has a unique maximal ideal. Prove that if R is a local ring with maximal ideal M then every element of (R-M) is a unit. Prove conversely that if R is a commutative ring with 1 in which the set of non-units forms an ideal M, then R is a local ring with unique maximal ideal M.

### Commutative Rings and Nilpotent Elements

Let R be a commutative ring with 1 not equal to zero. Prove that if "a" is a nilpotent element of R then 1-ab is a unit for all "b" in R.

### Irreducible Elements, Additive Abelian Groups, Quotient Rings and Chinese Remainder Theorem

Please see the attached file for the fully formatted problems.

### Sqrarefree Integers, Fields, Conductors and Maximal Ideals

Let D be a squarefree integer, and let 0 be the ring of integers in the quadratic field Q(sqrtD). For any positive integer f prove that the set Of = Z[fw] = {a + bfw | a, b E Z} is a subring of 0 containing the identity. Prove that [O:Of]= f (index as additive abelian groups). Prove conversely that a subring of 0 containing the

### Class of all finite unions of sets of the form A×B

Topology Sets and Functions (XLIII) Functions Let X and Y be non-empty sets and let A and B be rings of subsets of X and Y respectively. Show that the class of all finite u

### Short Exact Sequences

Let R be a subring of a ring S and let 0 -> M -> N -> P -> 0 be a short exact sequence of S-modules. Prove or disprove the following statements: (i) If the sequence is split over S, then it is split over R. (ii) If the sequence is split over R, then it is split over S. See the attached file.

### Ring theory question about polynomial rings

Please help with the following problem. Let R={f(x) included in Q[x]: the coefficient of x in f(x) is 0} Prove that R is a subring of Q[x].

### Ring Theory Problem

If R is a ring and p(x) is included in R[x] then f(x) is the associated polynomial function from R to R. Find a p(x) included in Zmod2[x] such that f(x)=0 for all x included in zmod2. I know that Zmod2 is all the polynomials whose coefficients are 0 and 1 but I have no idea what I am I trying to look for.

### Commutative Rings, Homomorphisms and Ideals

Show that if R and S are commutative rings with 1, phi:R-->S is a homomorphism of R onto S, and I is an ideal of R, then phi[I]={phi(r): r included in I} is an ideal of S.

### If R is an integral domain, then so is R[x_1, x_2, ... , x_n] . Or, Prove that if R is an integral domain, then R[x_1, x_2, ... , x_n] is also an integral domain.

Modern Algebra Ring Theory (L) Polynomial Rings over Commutative Rings Unit Element Integral Domain

### If R is an Integral Domain, then so is R[x].

If R is an integral domain, then so is R[x]. Prove that if R is an integral domain, then R[x] is also an integral domain. See attached file for full problem description.

### If R is a unique factorization domain and if a,b are in R, then a and b have a least common multiple (l.c.m.) in R.

If R is a unique factorization domain and if a,b are in R, then a and b have a least common multiple (l.c.m.) in R. See the attached file.