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Ring Theory

Ideals and Factor Rings : Locality

Problem: A ring R is called a local ring if the set J(R) of nonunits in R forms an ideal. If p is a prime, show that Z(p) = {n/m belonging to Q | p does not divide m } is local. Describe J(Z(p))

Ideals and Factor Rings : Annihilator

Problem: If X is contained in R is a nonempty subset of a commutative ring R, define the annihilator of X by ann(X) = { a belonging to R | ax=0 for all x belonging to X} Show that X is contiained in ann[ann(X)] AND Show that ann(X) = ann{ann[ann(X)]}

Ideals and Factor Rings

Problem: Note: | | is trying to denote a matrix If R = |S S| |0 S| and A = |0 S| |0 0| , S and ring, show that A is an ideal of R and describe the cosets in R/A

Rings and Units

Given r and s in a ring R, show that 1 + rs is a unit if and only if 1 + sr is a unit.

Quotient Ring

Please see the attached PDF file. I would prefer a solution in PDF format. Thanks!

Rings : Elements

Please see the attached file for the fully formatted problems. Describe the ring obtained from Z12 by adjoining the element 1/2 (the inverse of 2).

Rings : Fields

Please see the attached file for the fully formatted problems. Prove that Z5/(x2 + x + 1) is a field. How many elements are there in this field? Can you also represent it as Z5[x]/(x2-a) where a is some element of Z5?

Describe the Rings

Please see the attached file for the fully formatted problem. Describe the rings: Z[x]/(x2 − 3, 2x + 4), Z[i]/(2 + i) where i2 = −1.