Question

D. Let A, B be two rings. 1. If L is an ideal of A × B, prove that there exists an ideal I of A and an ideal J of B such that L = I × J. 2. With the notation above, prove that L is a maximal ideal of A × B if and only if either I = A and J is maximal, or else I is maximal and J = B.

          D. Let A, B be two rings.
1. If L is an ideal of A × B, prove that there exists an ideal I of A and an ideal J of B such that L = I × J.
2. With the notation above, prove that L is a maximal ideal of A × B if and only if either I = A and J is maximal, or else I is maximal and J = B.

D. Let A, B be two rings.
1. If L is an ideal of A × B, prove that there exists an ideal I of A and an ideal J of B such that L = I × J.
2. With the notation above, prove that L is a maximal ideal of A × B if and only if either I = A and J is maximal, or else I is maximal and J = B.

Added by Jonathon C.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Sri K

11/05/2022

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Step 1: Assume L is an ideal of A * B. Show more…

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Texts: D. Let A, B be two rings. 1. If L is an ideal of A * B, prove that there exists an ideal I of A and an ideal J of B such that L = I * J. 2. With the notation above, prove that L is a maximal ideal of A * B if and only if either I = A and J is maximal, or else I is maximal and J = B.