query: Artinian affine $K$-algebra
positive_passages.docid: 79460
positive_passages.text: Atiyah-Macdonald, Exercise 8.3: Artinian iff finite k-algebra.
negative_passages.docid: 150911
negative_passages.text: Prove that the ring is Artinian and find a composition series of ideals
negative_passages.docid: 15626
negative_passages.text: Relations of a~b iff b = ak^2
negative_passages.docid: 63541
negative_passages.text: Projective and injective modules; direct sums and products
negative_passages.docid: 126295
negative_passages.text: What is an $R$-algebra?
negative_passages.docid: 121100
negative_passages.text: Rootspaces are $\mathop{ad}$ nilpotent
negative_passages.docid: 63192
negative_passages.text: A chain ring with Krull dimension greater than one
negative_passages.docid: 44065
negative_passages.text: If $B$ is finitely generated as a $k$-algebra, and $\phi:A\to B$ is a $k$-algebra map, is $\phi^{-1}(M)$ maximal for any maximal $M\subset B$?
negative_passages.docid: 13606
negative_passages.text: Prove that $K$ is isomorphic to a subfield of the ring of $n\times n$ matrices over $F$.
negative_passages.docid: 66546
negative_passages.text: Tensor product of reduced $k$-algebras must be reduced?
negative_passages.docid: 83766
negative_passages.text: If $B$ is a graded $A$-algebra, then $\operatorname{Proj}B$ is an $A$-scheme
negative_passages.docid: 27096
negative_passages.text: Hilbert polynomial of an hypersurface in projective space
negative_passages.docid: 50656
negative_passages.text: Intuition behind $k$-algebra, $k$-algebra morphisms?
negative_passages.docid: 33063
negative_passages.text: Are all simple left modules over a simple left artinian ring isomorphic?
negative_passages.docid: 77137
negative_passages.text: The rank of Jacobian matrix at a point of affine variety is independent of choice of generators
negative_passages.docid: 136654
negative_passages.text: $(x_1-a_1, x_2-a_2)$ is a maximal ideal of $K[x_1,x_2]$
negative_passages.docid: 78298
negative_passages.text: On the nilpotence of the matrix $AB-BA$