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Ring Theory

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

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

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.

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.

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