Nilpotent Jacobson Radical Research Articles

Classical Radicals of Invariants of Algebraic Skew Derivations

Let [Formula: see text] be an automorphism and[Formula: see text] be a [Formula: see text]-skew [Formula: see text]-derivation of an [Formula: see text]-algebra [Formula: see text]. We prove that if [Formula: see text] is semiprimitive and [Formula: see text] is algebraic, then the subalgebra [Formula: see text] has nilpotent Jacobson radical. Using this result, we obtain similar relations for the Baer prime radical, the Levitzki locally nilpotent radical, and the Köthe nil radical when the field [Formula: see text] is uncountable. Then we apply it to actions of the [Formula: see text]-dimensional Taft Hopf algebra [Formula: see text] and the [Formula: see text]-analogue [Formula: see text] of the enveloping algebra of the Lie algebra [Formula: see text].

Locally nilpotent skew extensions of rings

We extend existing results on locally nilpotent differential polynomial rings to skew extensions of rings. We prove that if G={σt}t∈T is a locally finite family of automorphisms of an algebra R, D={δt}t∈T is a family of skew derivations of R such that the prime radical P of R is strongly invariant under D, then the ideal P〈T,G,D〉⁎ of R〈T,G,D〉, generated by P, is locally nilpotent. We then apply this result to algebras with locally nilpotent derivations. We prove that any algebra R over a field of characteristic 0, having a surjective locally nilpotent derivation d with commutative kernel, and such that R is generated by ker⁡d2, has a locally nilpotent Jacobson radical.

Про спадково незвiднi унiмономiальнi зображення циклiчних p-груп над локальними кiльцями характеристики p

The task of the description up to equivalency of the matrix representations of finite pgroups of order greater then p aver a commutative local ring of characteristics of ps (s > 0) that is not a field contains the classical unsolved problem of pair of matrices over a field. Therefore, consideration of partial cases and the study of special representation matrix representations is importent. Let K be a commutative ring with a identity, G is a finitely generated group with some fixed system of generator elements a1,...,ar. Every matrix representation of the group G over the ring K equivalent to Γ : ai → E+Mi (i = 1,...,r), where E be an identity n×n-matrix, Mi be a monomial n×n-matrix (i = 1,...,r), we call unimonomial. An reducible unimonomial representation over the ring K, in a reducible form of which on diagonal blocks at least one unimonomial representation is formed, we call hereditary reducible over the ring K. A series of unimonomial representations of a finite cyclic p-group H = haiover commutative local principle ideal ring K of characteristic p with nilpotent Jacobson radical of the degree l (1 < l < ∞) of the form a → E + 0 ... 0 1 1 ... 0 0 . . . ... . . . . . . 0 ... 1 0!· diag[ε1ts1,...,εntsn], where si ≥ 0, εi be elements from K∗ (i = 1,...,n), t be an generator element of Jacobson radical ring K. It is making up clear the criterion, when the map of the given form sets the representation of the group H (P|H|−1 j=0 si+j ≥ l (i = 1,...,n), herethe indexes are considered by the module n). It have been found the sufficient condition of hereditary irreducibility of the constructed representations ((Pn i=1 si,n) = 1,tPn i=1 si 6= 0). In addition, we can obtain the hereditary reducibility of the constructed representations in the case when (Pn i=1 si,n) > 1,Qn i=1 εi = 1). Based on the researches of Bondarenko V. M., Bortosh M. Yu. of similarity of the monomial matrices it is making up clear the criterion the equivalence of the constructed representations (the corresponding sequences (s1,...,sn) are cyclically equivalent a relevant productsQn i=1 εi are equal modulo Ann (ts), where s is the largest member of the weight sequence (s1,...,sn)). In the case of the finiteness of the ring K by computation in the GAP system it have been found the number of all, up to equivalence, constructed unimonomial hereditary irreducible matrix representations of a cyclic nontrivial p-group depending on the number of elements of the residue class field of the ring K.

Semiartinian Profinite Algebras have Nilpotent Jacobson Radical

We give a method to study the finiteness of the coradical filtration of a coalgebra; as a consequence, we show that a left semiartinian profinite algebra has nilpotent Jacobson radical and is right semiartinian too. Equivalently, we show that a for a semilocal profinite algebra, T-nilpotence implies nilpotence for the Jacobson radical. This answers two open questions from Iovanov et al. (J Algebra 320(5):2144–2155, 2008).

The prime spectrum of algebras of quadratic growth

We study prime algebras of quadratic growth. Our first result is that if A is a prime monomial algebra of quadratic growth then A has finitely many prime ideals P such that A / P has GK dimension one. This shows that prime monomial algebras of quadratic growth have bounded matrix images. We next show that a prime graded algebra of quadratic growth has the property that the intersection of the non-zero prime ideals P such that A / P has GK dimension 2 is non-zero, provided there is at least one such ideal. From this we conclude that a prime monomial algebra of quadratic growth is either primitive or has non-zero locally nilpotent Jacobson radical. Finally, we show that there exists a prime monomial algebra A of GK dimension two with unbounded matrix images and thus the quadratic growth hypothesis is necessary to conclude that there are only finitely many prime ideals such that A / P has GK dimension 1.

Structures and Representations of Generalized Path Algebras

It is shown that an algebra $\Lambda $ can be lifted with nilpotent Jacobson radical $r = r(\Lambda)$ and has a generalized matrix unit $\{e_{ii}\}_I$ with each $\bar e_{ii} $ in the center of $\bar \Lambda = \Lambda /r$ iff $\Lambda $ is isomorphic to a generalized path algebra with weak relations. Representations of the generalized path algebras are given. As a corollary, $\Lambda $ is a finite algebra with non-zero unity element over perfect field $k$ (e.g. a field with characteristic zero or a finite field) iff $\Lambda $ is isomorphic to a generalized path algebra $k (D, \Omega, \rho)$ of finite directed graph with weak relations and $dim {\} \Omega < \infty $; $\Lambda $ is a generalized elementary algebra which can be lifted with nilpotent Jacobson radical and has a complete set of pairwise orthogonal idempotents iff $\Lambda $ is isomorphic to a path algebra with relations.

Linearly compact rings having a simple group of units

Those linearly compact rings with identity having a simple group of units and a transfinitely nilpotent Jacobson radical are identified. A consequence of this characterization is Cohen and Koh's classification theorem for compact rings with identity having a simple group of units.

Complex Representations of Matrix Semigroups

Let M be a finite monoid of Lie type (these are the finite analogues of linear algebraic monoids) with group of units G. The multiplicative semigroup .4 (F) , where F is a finite field, is a particular example. Using HarishChandra's theory of cuspidal representations of finite groups of Lie type, we show that every complex representation of M is completely reducible. Using this we characterize the representations of G extending to irreducible representations of M as being those induced from the irreducible representations of certain parabolic subgroups of G. We go on to show that if F is any field and S any multiplicative subsemigroup of .4 (F) , then the semigroup algebra of S over any field of characteristic zero has nilpotent Jacobson radical. If S = .4 (F) , then this algebra is Jacobson semisimple. Finally we show that the semigroup algebra of .4 (F) over a field of characteristic zero is regular if and only if ch(F) = p > 0 and F is algebraic over its prime field.

Complex representations of matrix semigroups

Let M M be a finite monoid of Lie type (these are the finite analogues of linear algebraic monoids) with group of units G G . The multiplicative semigroup M n ( F ) {\mathcal {M}_n}(F) , where F F is a finite field, is a particular example. Using Harish-Chandra’s theory of cuspidal representations of finite groups of Lie type, we show that every complex representation of M M is completely reducible. Using this we characterize the representations of G G extending to irreducible representations of M M as being those induced from the irreducible representations of certain parabolic subgroups of G G . We go on to show that if F F is any field and S S any multiplicative subsemigroup of M n ( F ) {\mathcal {M}_n}(F) , then the semigroup algebra of S S over any field of characteristic zero has nilpotent Jacobson radical. If S = M n ( F ) S = {\mathcal {M}_n}(F) , then this algebra is Jacobson semisimple. Finally we show that the semigroup algebra of M n ( F ) {\mathcal {M}_n}(F) over a field of characteristic zero is regular if and only if ch ⁡ ( F ) = p &gt; 0 \operatorname {ch} (F) = p &gt; 0 and F F is algebraic over its prime field.

QE rings in characteristic pn

AbstractWe show that all QE rings of prime power characteristic are constructed in a straightforward way out of three components: a filtered Boolean power of a finite field, a nilpotent Jacobson radical, and the ring Zp. or the Witt ring W2(F4) (which is the characteristic four analogue of the Galois field with four elements).

Certain Artinian Rings are Noetherian

Throughout this paper the word “ring” will mean an associative ring which need not have an identity element. There are Artinian rings which are not Noetherian, for example C(p∞) with zero multiplication. These are the only such rings in that an Artinian ring R is Noetherian if and only if R contains no subgroups of type C(p∞) [1, p. 285]. However, a certain class of Artinian rings is Noetherian. A famous theorem of C. Hopkins states that an Artinian ring with an identity element is Noetherian [3, p. 69]. The proofs of these theorems involve the method of “factoring through the nilpotent Jacobson radical of the ring”. In this paper we state necessary and sufficient conditions for an Artinian ring (and an Artinian module) to be Noetherian. Our proof avoids the concept of the Jacobson radical and depends primarily upon the concept of the length of a composition series. As a corollary we obtain the result of Hopkins.