**ERIC Number:**EJ781082

**Record Type:**Journal

**Publication Date:**2008-Jan

**Pages:**6

**Abstractor:**Author

**Reference Count:**3

**ISBN:**N/A

**ISSN:**ISSN-0020-739X

Four Proofs of the Converse of the Chinese Remainder Theorem

Dobbs, D. E.

International Journal of Mathematical Education in Science and Technology, v39 n1 p104-109 Jan 2008

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 [double-struck z][subscript m] [direct sum] [double-struck z][subscript n] and [double-struck z][subscript mn] 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.

Descriptors: Mathematical Logic, Algebra, Validity, Numeracy, Number Concepts, Generalization, Logical Thinking, Mathematics Instruction, Demonstrations (Educational)

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**Publication Type:**Journal Articles; Reports - Descriptive

**Education Level:**N/A

**Audience:**N/A

**Language:**English

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