Frobenius polycyclic ring
Frobenius multicomponent ring(Frobenius, Arrowhead: Frobenius algebra), orFrobenius algebra Is a thing with special in giving a good dual theory among the studies.
Frobenius multiple circle began to be studied as a generalization by and in the 1930's, and it was named after. () And especially () For the first time discovered a rich dual theory. Using this, we characterize Frobenius multicomponent ring in (), and this property of Frobenius multicomponent ring is perfect I called it duality. Frobenius multiway ring was generalized to. This is right. Recently, the interest in Frobenius multicircle has been rising from the connection with.
⇧ Frobenius multiple circle ⊃ symmetric multiple ring ⊃ ⊃ ⊃
table of contents
Definition
A on * k isFrobenius multiple circleis σ : _ A _ x _ A _ _ _ _ _
σ _ ( ab , _ c ) = σ _ (_a , bc )
It means that there are things that satisfy. This bilinear format is calledFrobenius format(Frobenius form).
Equivalent characterization refers to linear mapping λ : _ A _ → _ k _ and that does not include left nonzero (λ ).
When Frobenius type σ is Frobenius polygon, or when the same condition λ _ ( ab ) = _ λ _ ( ba _) is satisfied, It is called symmetric multidimensional ring.
There are also different concepts that are almost irrelevant.
An example
1. Frobenius multivariate with Frobenius form σ _ ( a , _ b ) = ( ab ) on the body k. 2. Arbitrary Finite Dimension Unit Coupled Multirings A _ has its own homomorphism to its own homomorphic ring End (_A ). The bilinear form can be defined on A as in Example 1. If this bilinear form is nondegenerate, A _ has the structure of Frobenius multicircle. 3. All on the body is the Frobenius multidimensional ring with the following Frobenius form _σ , ie σ _ ( a , _ b ) is the factor of the identity element in _ a _ _ . This is the special case of Example 2. 4. For body _k, 4dimensional k  algebra _ k _ [ x , _ y ] / ( x _ 2, _ y _ 2) are Frobenius multicomponent rings. This follows the characterization of the exchange Frobenius ring described below. This ring is a local ring with the ideal generated with x _ and _ y _ as the local idea because it has the only minimal ideal generated by _ x _ . 5. For _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ is not a Frobenius multi  dimensional**. From xA _ _ derived from x ↦ y {\ displaystyle x \ mapsto y}! [x \ mapsto y] (https://wikimedia.org/api/rest_v1/media/math/render/svg/eb2452f2d32e5424f3db361de033fd49a73f9dcc) Of _A homomorphism can not be extended to A homomorphism from A _ _ _, hence the ring is not self  primitive and not Frobenius.
nature
 Frobenius multidimensional ring and / or are Frobenius multiway ring.
 It is the same that having a finite dimensional multiple ring on the body is Frobenius, that the right is introductory and that the plural ring has unique.
 Convertible locality Frobenius polygon is just like being local and including finite dimensions on the remainder.
 Frobenius multiple ring is (), in particular, it is left (right) and left (right).
 For body k, it is equivalent that the finite dimensional unitary coupled multiway ring is Frobenius and that the imported right_A_  group Hom _ k _ ( A , _ k ) is isomorphic to the right of _ A .
 For infinite field k, the finite dimensional unitary coupled multiway ring is Frobenius unless there are only a finite number of local minima. If _ _ is a finite order of _ _, finite dimensions _ F _  multiple rings are naturally finite dimensions _ k _  multiple rings by, it is equivalent to Frobenius _ F _  multicircular ring and Frobenius _k _  multiring . In other words, the Frobenius nature does not depend on the body unless the multicomponent ring is a finite dimensional multiple ring.
 Similarly, if F is a finite order extension field of k, all k  multiple rings _ A _ naturally generate F _  multiply ring _ F _ ⊗ _ k _ A _ _ is Frobenius k _  multicircular ring and _ F _ ⊗ It is equivalent that _k _ A _ is Frobenius _ F _  multicomponent ring.
 In the unitary coupling multiple rings of right finite dimension expression, the Frobenius plural circle A _ is exactly like a multinary circle whose _M _ has the same dimensions as its _ A _ dual Hom _ A _ (_M , _ A ) is there. Among these multiple rings, the _A dual of simple additions is always simple.
footnote
1. ****, Definition 4.2.5.
References
; (1937),, Proc. Nat. Acad. Sci. USA 23(4): 236240,:,,, DeMeyer, F., Ingraham, E. (1971), Separable Algebras over Commutative Rings , Lect. Notes Math 181, Springer (1958), "Remarks on quasiFrobenius rings", Illinois Journal of Mathematics 2: 346354,, (1903), "Theorie der hyperkomplexen Größen I" (German), Sitzungsberichte der Preussischen Akademie der Wissenschaften : 504537, Kock, Joachim (2003), Frobenius Algebras and 2D Topological Quantum Field Theories, London Mathematical Society student texts, Cambridge: Cambridge University Press, Lam, T. Y. (1999),, Graduate Texts in Mathematics No. 189, Berlin, New York: Lurie, Jacob,, (1939),, (Annals of Mathematics)40(3): 611633,::,,, (1941),, __ (Annals of Mathematics)42(1): 121,:,,, (1938),, 39(3): 634658,::,,,, Onodera, T. (1964), "Some studies on projective Frobenius extensions", __18: 89107 Weibel, Charles A. (1994).. Cambridge University Press.
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