In Atiyah-MacDonalds book on Commutative Algebra we have on page 6 the following statement ($\mathfrak{a}, \mathfrak{b}$ denote ideals in a ring):

"(...) in $\mathbf{Z}$ we have $(\mathfrak{a} + \mathfrak{b}) (\mathfrak{a} \cap \mathfrak{b}) = \mathfrak{a} \mathfrak{b}$; but in general we have only $(\mathfrak{a} + \mathfrak{b}) (\mathfrak{a} \cap \mathfrak{b}) \subseteq \mathfrak{a} \mathfrak{b}$."

Unfortunately they do not give any example where the inclusion is strict. Does anyone know such an example?

Thanks a lot!


In the ring of real bivariate polynomials consider the two ideals generated by the monomials x and y. Then the left hand side contains no polynomials of degree less than 3 but the right hand side contains xy.

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