
Ideals : Cyclic Module in a Commutative Ring
We can see f is an additive homomorphism of groups.

Rings of Unity, Monoid, Momomorphism and Invertible Elements
Since and , then . And we have . Thus is the inverse of . So is a unit of . Rings of Unity, Monoid, Momomorphism and Invertible Elements are investigated. The solution is detailed and well presented.

Modules, Polynomial Rings, Finite Dimensional Vector Space and Multiplication
There is a theorem (see Fraleigh or any god text) that *all* finitely generated abelian groups are isomorphic to products of Z mod n, in fact of Z mod p^m where p is prime. Therefore, our M here cannot be a finitely generated group.

Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields.
Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields

the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields.
Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields

Facts About Group Theory
There are finite and infinite cyclic groups. All infinite cyclic groups are isomorphic to the additive group of integers. All cyclic groups of an order n are isomorphic to the additive group of integers Z_n modulo n .

Rings
231207 Rings: Addition and Multiplication Which of the following are rings with respect to the usual definition of addition and multiplication?

Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields.
Topics usually include the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields

Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings, primitive roots, quadratic reciprocity and quadratic fields.
This solution is comprised of a detailed explanation to answer questions including the Euclidean algorithm, primes and unique factorization, congruences, Chinese Remainder Theorem, Hensel's Lemma, Diophantine equations, arithmetic in polynomial rings,