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Results tagged with ra.rings-and-algebras
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user 430
Non-commutative rings and algebras, non-associative algebras, universal algebra and lattice theory, linear algebra, semigroups. For questions specific to commutative algebra (that is, rings that are assumed both associative and commutative), rather use the tag ac.commutative-algebra.
34
votes
Accepted
Lie algebra semisimple if and only if perfect?
No. A Lie algebra satisfying that property is called perfect. For an example of a perfect Lie algebra that isn't semisimple, take a semisimple $L$ and a nontrivial irreducible representation $V$ of $L …
- 10.3k
8
votes
Accepted
What is an example of a ring in which the intersection of all maximal two-sided ideals is no...
Certainly every maximal ideal is the annihilator of a simple R-module, but the converse isn't true. See Exercise 4.8 in Lam's "Exercises in Classical Ring Theory" for an example.
- 10.3k
29
votes
Accepted
Infinite-dimensional normed division algebras
A MathSciNet search reveals a paper by Urbanik and Wright (Absolute-valued algebras. Proc. Amer. Math. Soc. 11 (1960), 861–866) where it is proved that an arbitrary real normed algebra (with unit) is …
- 10.3k
7
votes
Accepted
Which linear transformations between f.d. Hilbert spaces contract the inner product?
Such a map will preserve orthogonality, and any such map must be a scalar multiple of an isometry. This is true in great generality, e.g. the map $T$ doesn't have to be linear, and $U$ and $V$ don't h …
- 10.3k
5
votes
Are there any nontrivial ring homomorphisms $M_{n+1}(R)\rightarrow M_n(R)$?
We can also rule out the case of commutative $R$ by appealing to the Artin–Procesi theorem: an Azumaya algebra of constant rank $(n+1)^2$ (e.g. $M_{n+1}(R)$) satisfies all the $\mathbb Z$-multilinear …
- 10.3k