However, even over the ring of integers, the module of rational numbers has an endomorphism ring that is a division ring, specifically the field of rational numbers.
22.
It is not only an abelian group but is also a ring with multiplication given by function composition; it is called the endomorphism ring of " M ".
23.
However, this is a stricter notion, since the endomorphism ring of a countably infinite dimensional vector space is semiprimitive, but not a subdirect product of simple rings,.
24.
More generally, square roots can be considered in any context in which a notion of " squaring " of some mathematical objects is defined ( including algebras of matrices, endomorphism rings, etc .)
25.
The possible types of endomorphism ring have been classified, as rings with lattice ? in "'C " "'d ", one must take into account the Riemann relations of abelian variety theory.
26.
The endomorphism ring of this is an order in a quaternion algebra over the rationals, and should act on the first cohomology group, which should be a 2-dimensional vector space over the coefficient field by analogy with the case of a complex elliptic curve.
27.
20 ) " The endomorphism ring of an Abelian group is trivial if and only if the Abelian group in question is the trivial group . " Explanation / reference ? ( It's easy : there is always the identity endomorphism and the zero endomorphism .)
28.
It is known that the endomorphism ring End R ( " M " ) is a semilocal ring which is very close to a local ring in the sense that End R ( " M " ) has at most two maximal right ideals.
29.
There are " three types " of irreducible real representations of a finite group on a real vector space " V ", as the endomorphism ring commuting with the group action can be isomorphic to either the real numbers, or the complex numbers, or the quaternions.
30.
If A | B is in addition to being depth two a Frobenius algebra extension, the right and left endomorphism rings are anti-isomorphic, which restricts to an antipode on the bialgebroid \ mbox { End } \, { } _ BA _ B satisfying axioms of a Hopf algebroid.
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