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In a graded ring, a homogeneous ideal is an ideal which is generated by homogeneous elements. This seems straightforward enough, but if your ideal is presented abstractly instead of in terms of generators, it becomes less obvious how to verify homogeneity. In this talk, we will discuss several different techniques for showing an ideal is homogeneous, as well as comparing the strengths and limitations of these techniques. We'll illustrate these via proving useful results on how homogeneity is preserved, such as showing that associated primes of homogeneous ideals are homogeneous ideals, and showing the integral closure of a homogeneous ideal is homogeneous.

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