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Ring Theory

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

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.

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?

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.

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

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

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