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# Ring Theory

### 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.

### Properties of Elements of a Ring

Give an example of two elements a,b in a ring R such that a(b)=0 but b(a) <> 0. See attached file for full problem description. keywords: property

### Prove properties of a ring with additive identity 0.

Let R be a ring with additive identity 0. Prove the following: (a) For all a in R, a(0) = 0. (b) a(-b)=-(ab). NOTE: see attached word document for clearer notations.

### 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

### Ring Unity

Let R be a ring with unity 1 and let S be a subring of R. Is it possible that S has unity e such that e does not equal 1?

### Subrings and Division Rings

Prove that any finite subring of a division ring is a division ring.

### Important information about Rings : Ideals

Let S be a subset of a set X. Let R be the ring of real-valued functions on X, and let I be the set of real-valued functions on X whose restriction to S is zero. Show that I is an ideal in R.

### 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

### Commutative Rings, Subrings and Submodules

Problem: I need to show that (i) leads to (ii), then (ii) leads to (iii): Let S be a commutative ring, R be a subring in S and x be an element from S. Show that the following are equivalent: (i) There exist from R where such that ; In other words x is a root of normalized polynomial over R. (ii) Submodule R - modu

### F[x]/(f(x)) is a field with p^n elements.

Modern Algebra Ring Theory (XXXII) Polynomial Ring Irreducible Polynomial If f(x) is in F[x], where F is the field

### Euclidean Domains and Principal Integral Domains

Determine whether the ring R=Z[(1+sqrt-5)/2] is a Euclidean Domain. And whether the ring R=Z[(1+ sqrt-5)/2] is a Principal Ideal Domain See attached file for full problem description.

### Complex Numbers, Integer and Rational Coefficient

Consider the ring Z[i] of complex numbers with integer coefficients. Prove or disprove that the field of fractions associated with Z[i] is isomorphic to Q[i], the field of complex numbers with rational coefficients. keywords: isomorphisms

### F[x]/(x^2 + 1) is isomorphic to the field of complex numbers

Modern Algebra Ring Theory (XXXI) Polynomial Ring Irreducible Polynomial Let F be a field of real nu

### Polynomial Rings: Irreducible Polynomials

Prove that x^2 + 1 is irreducible over the field F of integers mod 11 and prove directly that F[x]/(x^2 + 1) is a field having 121 elements.

### Ring Isomorphisms Functions

Prove that Z[X] and Q[X] are not isomorphic.

### 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.

### Rings and Ideals

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.

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

Let p be a prime and let G be an abelian group of order P^n . Prove that the nilradical of the group ring FPG is the augmentation ideal. Please see the attached file for the fully formatted problem.

### Commutative Groups and Augmentation Ideals

Let G={g1,..., gn } be a finite group and assume R is commutative. Prove that if r us any element of the augmentation ideal of RG then r(g1+......gn)=0

### Ring Homomorphisms and Ideals

Let : R->Q be a ring homomorphism , and suppose that I is a non-trivial ideal of R. Prove or disprove that (I)={ (i)| i I } is necessary an ideal of Q. Let : R->Q be a ONTO ring homomorphism , and suppose that I is a non-trivial ideal of R. Prove or disprove that (I)={ (i)| i I } is necessary an ideal of Q. Plea