Chinese Remainder Theorem - Non-commutative Case: A Counter-example

Non-commutative Case: A Counter-example

The Chinese remainder theorem does not hold in the non-commutative case. Consider the ring R of non-commutative real polynomials in x and y. Let I be the principal two-sided ideal generated by x and J the principal two-sided ideal generated by . Then but .

Read more about this topic:  Chinese Remainder Theorem