I'm not quite sure of what you are asking. I guess that 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.

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.