Resumen de Four proofs on the converse of the Chinese Remainder Theorem
Four proofs, designed for classroom use in varying levels of courses on abstract algebra, are given for the converse of the classical Chinese Remainder Theorem over the integers. In other words, it is proved that if m and n are integers greater than 1 such that the abelian groups [image omitted] and [image omitted] are isomorphic, then m and n are relatively prime. The first proof, which involves the order of an element in a finite group, can be given early in a first course on abstract algebra. The second proof, which can be given later in such a course, depends on Lagrange's Theorem and the Second Isomorphism Theorem. The final two proofs, which involve tensor products and exterior powers, can be given in more advanced courses that develop or assume multilinear algebra. The latter two proofs generalize to give the converse of the Chinese Remainder Theorem for modules over commutative rings.Four proofs, designed for classroom use in varying levels of courses on abstract algebra, are given for the converse of the classical Chinese Remainder Theorem over the integers. In other words, it is proved that if m and n are integers greater than 1 such that the abelian groups [image omitted] and [image omitted] are isomorphic, then m and n are relatively prime. The first proof, which involves the order of an element in a finite group, can be given early in a first course on abstract algebra. The second proof, which can be given later in such a course, depends on Lagrange's Theorem and the Second Isomorphism Theorem. The final two proofs, which involve tensor products and exterior powers, can be given in more advanced courses that develop or assume multilinear algebra. The latter two proofs generalize to give the converse of the Chinese Remainder Theorem for modules over commutative rings.Four proofs, designed for classroom use in varying levels of courses on abstract algebra, are given for the converse of the classical Chinese Remainder Theorem over the integers. In other words, it is proved that if m and n are integers greater than 1 such that the abelian groups [image omitted] and [image omitted] are isomorphic, then m and n are relatively prime. The first proof, which involves the order of an element in a finite group, can be given early in a first course on abstract algebra. The second proof, which can be given later in such a course, depends on Lagrange's Theorem and the Second Isomorphism Theorem. The final two proofs, which involve tensor products and exterior powers, can be given in more advanced courses that develop or assume multilinear algebra. The latter two proofs generalize to give the converse of the Chinese Remainder Theorem for modules over commutative rings.Four proofs, designed for classroom use in varying levels of courses on abstract algebra, are given for the converse of the classical Chinese Remainder Theorem over the integers. In other words, it is proved that if m and n are integers greater than 1 such that the abelian groups [image omitted] and [image omitted] are isomorphic, then m and n are relatively prime. The first proof, which involves the order of an element in a finite group, can be given early in a first course on abstract algebra. The second proof, which can be given later in such a course, depends on Lagrange's Theorem and the Second Isomorphism Theorem. The final two proofs, which involve tensor products and exterior powers, can be given in more advanced courses that develop or assume multilinear algebra. The latter two proofs generalize to give the converse of the Chinese Remainder Theorem for modules over commutative rings.
