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Jun 30 at 19:26 comment added Richard Stanley Hint: the socle of $I$ is contained in the socle of $A$.
Jun 30 at 19:21 comment added Chen Thank you! Why should any nonzero ideal intersect the socle non-trivially? Could you please give a hint/reference.
Jun 30 at 19:01 history answered Richard Stanley CC BY-SA 4.0