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Let $R$ be a domain with the quotient field $K$. Let $T$ be an overring of $R$ ($T$ contains $R$) and assume that $T$ is not equal to the localization of any multiplicative closed subset $S$ of $R$ ($T \neq R_{S}$ for any multiplicative closed subset of $R$). Let $A$ be a fractional ideal (not an integral ideal) of $R$. How can we show that $AT$ is a $T$-submodule of $K$?

This is obvious if $A$ is an integral ideal of $R,$ then $AT$ is an integral ideal of $T.$

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    I don't get what's your problem here. It's obvious that $AT$ is closed by the multiplications with elements of $T$, and the sum of two sums is a sum. – 2017-02-26

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