Let $K$ be a field and $A$ be a finite-dimensional $K$-algebra. Does $A$ have finitely many ideals? (I know that $A$ has finitely many prime ideals.)
2026-04-19 04:02:31.1776571351
Number of ideals in a finite-dimensional $K$-algebra
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Set $A=\mathbb C[X,Y]/(X^2,XY,Y^2)$. We have $\dim_{\mathbb C}A=3$, a basis is $\{1,x,y\}$, and the proper ideals of $A$ are exactly the $\mathbb C$-subspaces of the vector space generated by $x$ and $y$.