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.
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
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.
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.
Modern Algebra Ring Theory (XLI) Polynomial Rings over Commutative rings Integral Domain Unit Element
Modern Algebra Ring Theory (XL) Polynomial Rings over Commutative rings Integral Domain
Modern Algebra Ring Theory (XXXIX) Polynomial Rings over Commutative Rings Unit Element
Modern Algebra Ring Theory (XXXVIII) Polynomial Rings over Commutative Rings Integral Domain Degree of a Polyn
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?
Prove that any finite subring of a division ring is a division ring.
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.
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.
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
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
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
Modern Algebra Ring Theory (XXXVII) Polynomials over the Rational Field Monic Polynomial
If P is a prime number, prove that the polynomial x^n - p is irreducible over the rationals.
Modern Algebra Ring Theory (XXXIII) Polynomials over the Rational Field Euclidean Ring
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
Modern Algebra Ring Theory (XXXII) Polynomial Ring Irreducible Polynomial If f(x) is in F[x], where F is the field
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.
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
Modern Algebra Ring Theory (XXXI) Polynomial Ring Irreducible Polynomial Let F be a field of real nu
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.
Prove that Z[X] and Q[X] are not isomorphic.
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.
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.
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
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