Associated graded module

In mathematics, the associated graded ring of a ring R with respect to a proper ideal I is the graded ring:

Similarly, if M is a left R-module, then the associated graded module is the graded module over ${\displaystyle \operatorname {gr} _{I}R}$:

For a ring R and ideal I, multiplication in ${\displaystyle \operatorname {gr} _{I}R}$ is defined as follows: First, consider homogeneous elements ${\displaystyle a\in I^{i}/I^{i+1}}$ and ${\displaystyle b\in I^{j}/I^{j+1}}$ and suppose ${\displaystyle a'\in I^{i}}$ is a representative of a and ${\displaystyle b'\in I^{j}}$ is a representative of b. Then define ${\displaystyle ab}$ to be the equivalence class of ${\displaystyle a'b'}$ in ${\displaystyle I^{i+j}/I^{i+j+1}}$. Note that this is well-defined modulo ${\displaystyle I^{i+j+1}}$. Multiplication of inhomogeneous elements is defined by using the distributive property.

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