### 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

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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.

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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

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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

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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

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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

Find a commutative ring R with no ideal other than (0) and R that is not a Field.