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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

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).

### 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

### 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

### 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.

### 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].

### 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.

### Commutative ring with no non-zero nilpotent elements

Let R be a commutative ring with no non-zero nilpotent elements (that is, a^n = 0 implies a = 0). If f(x) = a_0 + a_1x + a_2x^2 +...+ a_mx^m in R[x] is a zero-divisor, prove that there is an element b is not equal to 0 in R such that ba_0 = ba_1 = ba_2 = ...=ba_m = 0. See attached file for full problem description.

### If R is an integral domain with unit element, prove that any unit in R[x] is a unit in R and any unit in R is also a unit in R[x].

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

### If R is an integral domain with unit element, prove that any unit in R[x] must already be a unit in R.

Modern Algebra Ring Theory (XL) Polynomial Rings over Commutative rings Integral Domain

### Prove that if R is a commutative ring with unit element then R[x] is also a commutative ring with unit element.

Modern Algebra Ring Theory (XXXIX) Polynomial Rings over Commutative Rings Unit Element

### deg (f(x)g(x)) = deg f(x) + deg g(x)

Modern Algebra Ring Theory (XXXVIII) Polynomial Rings over Commutative Rings Integral Domain Degree of a Polyn

### Rings and Principal Ideals: Left and Right Ideals

Show that if R is a ring with identity element and x is an element in R, then Rx = {rx: r in R} is the principal left ideal generated by x. Similarly, xR = {xr: r in R} is the principal right ideal generated by x.

### Show that a set of matrices is a ring without an identity element.

I am given the set of infinite-by-infinite matrices with real entries that have only finitely many nonzero entries. Also, the matrix has entries a_ij, where i and j are natural numbers and there is a natural number n such that a_ij = 0 if i >= n or j >= n. I need to understand how to show that this set of matrices is a ring w

### The Symmetric Difference of two Sets: A ring of sets is a non-empty class A of sets such that if A and B are in A, then A difference B and A intersection B are also in A. Show that A must also contain the A - B.

Topology Sets and Functions (XV) The Algebra of Sets Ring of Sets The Symmetric Difference of two Sets A ring of sets is a non-empty class A of sets

### The Symmetric Difference of two Sets: A ring of sets is a non-empty class A of sets such that if A and B are in A, then A difference B and A intersection B are also in A. Show that A must also contain the empty set.

Topology Sets and Functions (XIV) The Algebra of Sets Ring of Sets The Symmetric Difference of two Sets A ring of sets is a n

### Polynomials over the Rational Field: Monic Polynomial: If a is rational and x - a divides an integer monic polynomial, prove that a must be an integer.

Modern Algebra Ring Theory (XXXVII) Polynomials over the Rational Field Monic Polynomial

### Irreducible over the rationals.

If P is a prime number, prove that the polynomial x^n - p is irreducible over the rationals.

### Polynomials over the Rational Field: Euclidean Ring: Let D be a Euclidean ring, F its field of quotients. Prove the Gauss lemma for polynomials with coefficients in D factored as product of polynomials with coefficients in F.

Modern Algebra Ring Theory (XXXIII) Polynomials over the Rational Field Euclidean Ring