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I want to know that in the commutative rings, the definition of the primary module is coincide with the definition of the primary ring? And where i can find it? The primary ring is a ring when ab=0, so a=0 or a power of b is zero and a primary module is a module with only one associated prime.

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I'm not quite sure of what you are asking. I guess that what you want to check is that if $R$ is a primary ring, then it is primary as an $R$-module. In this case $R$ being primary means that if $I$ is the ideal of nilpotent elements, then $R/I$ is an integral domain. It follows that $I$ is the only associated prime and hence you get the definition you want.

I'm using Ernst Kunz Introduction to Commutative Algebra and Algebraic Geometry to get my definitions here.

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