On the Normal Subgroups of $G_2(A)$

Block-form control of linear time-invariant discrete-time systems defined over commutative rings

In this paper we prove a sandwich classification theorem for subgroups of the classical symplectic group over an arbitrary commutative ring $R$ that contain the elementary block-diagonal (or subsystem) subgroup $\operatorname{Ep}(ν, R)$ corresponding to a unitary equivalence realation $ν$ such that all self-conjugate equivalence classes of $ν$ are of size at least 4 and all not-self-conjugate classes of $ν$ are of size at least 5. Namely, given a subgroup $H$ of $\operatorname{Sp}(2n, R)$ such that $\operatorname{Ep}(ν, R) \le H$ we show that there exists a unique exact major form net of ideals $(σ, Γ)$ over $R$ such that $\operatorname{Ep}(σ, Γ) \le H \le N_{\operatorname{Sp}(2n,R)}(\operatorname{Sp}(σ, Γ))$. Further, we describe the normalizer $N_{\operatorname{Sp}(2n,R)}(\operatorname{Sp}(σ, Γ))$ in terms of congruences.

It was shown by C. Wickham [12] that “for a fixed positive integer g, there are finitely many isomorphism classes of finite commutative rings whose zero-divisor graph has genus g.” In this note, we give a short direct proof for this result. Moreover, we show that, if the zero-divisor graph of a commutative ring R has finite genus g, then either g = 0 or R is a finite ring. This immediately generalizes Wickham's theorem to arbitrary (not necessary finite) commutative rings.

This is a survey of results describing the parabolic subgroups of Chevalley groups over rings of various types. For Chevalley groups of classical types over an arbitrary commutative ring, a description of parabolic subgroups is obtained which may be considered definitive. Some errors in the results of V. M. Levchuk in Mat. Zametki, Volume 31, No. 4 (1982) are pointed out.

Resolutions of determinantal ideals: The submaximal minors

Given a certain factorization property of a ring [Formula: see text], we can ask if this property extends to the polynomial ring over [Formula: see text] or vice versa. For example, it is well known that [Formula: see text] is a unique factorization domain if and only if [Formula: see text] is a unique factorization domain. If [Formula: see text] is not a domain, this is no longer true. In this paper, we survey unique factorization in commutative rings with zero divisors, and characterize when a polynomial ring over an arbitrary commutative ring has unique factorization.

This paper is a continuation of RZhMat 1980, 5A439, where there was introduced the subgroup γ(δ) of the Chevalley group G(Φ,R) of type Φ over a commutative ring R that corresponds to a net δ, i.e., to a set б=(б∝),∝∈Φ, of ideals б∝ of R such that б∝бβ⊑б∝+β whenever α,Β,α+Β ∃Φ. It is proved that if the ring R is semilocal, then Γ(б) coincides with the group γ0δ considered earlier in RZhMat 1976, 10A151; 1977, 10A301; 1978, 6A476. For this purpose there is constructed a decomposition of γ(δ) into a product of unipotent subgroups and a torus. Analogous results are obtained for sub-radical nets over an arbitrary commutative ring.

Let M be a nonsingular quadratic module over a commutative ring R. The primary goal of this chapter is the proof of the following fact: If M is free of finite rank, then C(M) has a special element, or, equivalently, the Arf algebra A(M) of M is isomorphic to R[X](X2 - aX - b) for some a and b in R with a2 + 4b ∈ R*. This has the important consequence that the results developed in Chapters 7 and 8 for C(M) hold for any free M. In addition, we will establish that the discriminant a2 + 4b of X2 - aX - b is the classical discriminant of the quadratic module M (except for a factor 2 if rank M is odd). It was established in Chapter 9D that the Arf algebra A(M) is a separable quadratic algebra. The structural features of this algebra will play a decisive role in the proofs. Throughout, R is an arbitrary commutative ring.

We revisit a theorem of Grosshans and show that it holds over arbitrary commutative base ring k. One considers a split reductive group scheme G acting on a k-algebra A and leaving invariant a subalgebra R. Let U be the unipotent radical of a split Borel subgroup scheme. If R U = A U then the conclusion is that A is integral over R.

Due to the theoretical work and computer implementations of, for instance, Barakat, Quadrat and Robertz and their coauthors the theory of finitely generated (f.g.) modules over non-commutative regular noetherian rings of partial differential operators with variable coefficients like the Weyl algebras and over other similar rings has become constructive in recent years. In particular these authors compute the purity or grade filtration of a f.g. module by homological means and discuss its significance for the associated behavior. Pommaret and Quadrat noted this significance already in 1999. In this note it is shown that over an arbitrary commutative noetherian ring of operators the purity filtration of a finitely generated module can be easily computed by means of the the primary decomposition of its zero submodule and indeed, with smaller complexity, by inductively computing equidimensional parts. In most books on Constructive Commutative Algebra the connection between this primary decomposition, the equidimensional parts and the purity filtration is implicitly stated for cyclic modules. For many commutative rings of operators the standard signal modules are injective cogenerators. In this case the purity filtration of the module gives rise to a corresponding filtration of the dual behavior, and the primary decomposition induces additional sum representations of the pure dimensional factors of this filtration. For non-commutative rings of operators the standard signal modules are in general neither injective nor cogenerators, and for such signal modules the usefulness of the purity filtration of the module for the determination of the behavior and its structural properties is not obvious. It is also shown by a counter-example that dimensional purity of the module or behavior does not imply dimensional purity of the initial conditions according to Riquier of the associated homogeneous Cauchy problem.

In 1963, Gilmer characterized all finite commutative rings with a cyclic group of units and, in 1967, Eldridge and Fischer generalized these results to rings with minimum condition. In the present paper these results are extended to semiperf ect rings and generalizations of the three theorems are obtained. It is shown that a semiperfect ring with cyclic group of units is finite and is either commutative or is the direct sum of a commutative ring and the 2 x 2 upper triangular matrix ring over the field of two elements. Let R be semiperfect with an abelian group of units. It is shown that R is finite if either the group of units is finite or the group of units is finitely generated and the Jacobson radical is nil. The proofs of all these results depend on our main theorem: The structure of a semiperfect ring R with an abelian group of units is described completely up to the structure of commutative local rings. (That is commutative rings with a unique maximal ideal.) The groups of units of these local rings are shown to be direct factors of the group of units of R.

This chapter investigates the objects and mappings of algebraic geometry from a geometric point of view, making use especially of the algebraic tools of Chapter VII and of Sections 7–10 of Chapter VIII. In Sections 1–12, k denotes a fixed algebraically closed field. Sections 1–6 establish the definitions and elementary properties of varieties, maps between varieties, and dimension, all over k. Sections 1–3 concern varieties and dimension. Affine algebraic sets, affine varieties, and the Zariski topology on affine space are introduced in Section 1, and projective algebraic sets and projective varieties are introduced in Section 3. Section 2 defines the geometric dimension of an affine algebraic set, relating the notion to Krull dimension and transcendence degree. The actual context of Section 2 is a Noetherian topological space, the Zariski topology on affine space being an example. In such a space every closed subset is the finite union of irreducible closed subsets, and the union can be written in a certainway thatmakes the decomposition unique. Every nonempty closed set has a meaningful geometric dimension. In affine space the irreducible closed sets are the varieties, and each variety acquires a geometric dimension. The discussion in Section 2 applies in the context of projective space as well, and thus each projective variety acquires a geometric dimension. Moreover, any nonempty open subset of a Noetherian space is Noetherian. A nonempty open subset of an affine variety is called quasi-affine, and a nonempty open subset of a projective variety is called quasiprojective. Each quasi-affine variety or quasiprojective variety has a dimension equal to that of its closure, which is a variety. Sections 4–6 take up maps between varieties. Section 4 introduces spaces of scalar-valued functions on quasiprojective varieties–rational functions, functions regular at a point, and functions regular on an open set. The section goes on to relate these notions for the different kinds of varieties. Section 5 introduces morphisms, which are a restricted kind of function between varieties. The tools of Sections 4–5 together show that for many purposes all the different kinds of varieties can be treated as quasiprojective varieties. Section 6 introduces rational maps between varieties; these are not everywhere-defined functions, but each can be restricted to an open dense subset on which it is a morphism. Rational maps with dense image correspond to field mappings of the fields of rational functions, with the order of the mappings reversed. Section 7 concerns singularities at points of varieties, still over the field k. Zariski's Theorem was stated in Chapter VII for affine varieties and partly proved at that time. In the current context it has a meaning for any point of any quasiprojective variety. The section proves the full theorem, which characterizes singular points in a way that shows they remain singular under isomorphisms of varieties. Section 8 concerns classification questions over k for irreducible curves, i.e., quasiprojective varieties of dimension 1. From Section 6 it is known that two irreducible curves are equivalent under rational maps if and only if their fields of rational functions are isomorphic. The main theorem of Section 8 is that each such equivalence class of irreducible curves contains an everywhere nonsingular projective curve, and this curve is unique up to isomorphism of varieties. The points of this curve are parametrized by those discrete valuations of the underlying function field that are defined over k. Sections 9–12 relate the general theory of Sections 1–6 to the topic of solutions of simultaneous solutions of polynomial equations, as treated at length in Chapter VIII. Section 9 treats monomial ideals in k[X1, . . . , Xn], identifying their zero loci concretely and computing their dimension. The section goes on to introduce the affineHilbert function of this ideal, which measures the proportion of polynomials of degree≤ s not in the ideal. In the way that this function is defined, it is a polynomial for large s called the affine Hilbert polynomial of the ideal. Its degree equals the dimension of the zero locus of the ideal. Section 10 extends this theory from monomial ideals to all ideals, again concretely computing the dimension of the zero loci, obtaining an affine Hilbert polynomial, and showing that its degree equals the dimension of the zero locus of the ideal. Section 11 adapts the theory to homogeneous ideals and projective algebraic sets by making use of the cone in affine space over the set in projective space. Section 12 applies the theory of Section 11 to address the question how the dimension of a projective algebraic set is cut down when the set is intersected with a projective hypersurface. A consequence of the theory is the result that a homogeneous system of polynomial equations over an algebraically closed field with more unknowns than equations has a nonzero solution. Section 13 is a brief introduction to the theory of schemes, which extends the theory of varieties by replacing the underlying algebraically closed field by an arbitrary commutative ring with identity.

Multidimensional linear systems that are finite-dimensional over the coefficient field admit 1D-like state space realizations. We generalize this fact to the case where the base field is replaced by an arbitrary commutative Noetherian ring. It turns out that the concept of "state spaces" has to be replaced by that of "state modules", which are not necessarily free.

On the basis of combinatorial techniques of dihedral modules with ∞ \infty -simplicial faces, dihedral homology is constructed for involutive A ∞ A_{\infty } -algebras over arbitrary commutative unital rings.

In this paper we construct a (multihomogeneous) dualizing complex for A [Δ], the Stanley–Reisner ring of Δ over an arbitrary commutative noetherian ring A with identity, admitting a dualizing complex. In addition, some consequences are discussed.