A Non-Existence Theorem for Almost Split Sequences

Let k be a field, Q a quiver with countably many vertices and I an ideal of kQ such that kQ/I is a spectroid. In this note, we prove that there is no almost split sequence ending at an indecomposable not finitely presented representation of the bound quiver (Q, I). We then get that an indecomposable representation M of (Q, I) is the ending term of an almost split sequence if and only if it is finitely presented and not projective. The dual results are also true.     

Valuation ideals of order one in 2-dimensional regular local rings

Let υ be a prime divisor of a 2-dimensional regular local ring (R m) with algebraically closed residue field k. Zariski showed that a prime divisor υ of R is uniquely associated to a simple m-primary integrally closed ideal I of R, there exist finitely many simple υ-ideals including I, and all the other υ-ideals can be uniquely factored into products of simple υ-ideals. It is known that such an m-primary ideal I of R can be minimally generated by o(I) + 1 elements.Given a simple integrally closed ideal I of order one with arbitrary rank and its associated prime divisor υ, we find minimal generating sets of all the simple υ-ideals and describe factorizations of all the composite υ-ideals in terms of power products of simple υideals as explicitly as possible.     

M-Canonical ideals in integral domains

We prove that a Priifer domain R has an m-canonical ideal J, that is, an ideal I such that J: (I: J) = J for every ideal J of R, if and only if R is h-local with only finitely many maximal ideals that are not finitely generated; moreover, if these conditions are satisfied, then the product of the non-finitely generated maximal ideals is an m-canonical ideal of R     

Separable algebras over infinite fields are 2-generated and finitely presented

We prove that every separable algebra over an infinite field F admits a presentation with 2 generators and finitely many relations. In particular, this is true for finite direct sums of matrix algebras over F and for group algebras FG, where G is a finite group such that the order of G is invertible in F. We illustrate the usefulness of such presentations by using them to find a polynomial criterion to decide when 2 ordered pairs of 2 × 2 matrices (A, B) and (A′, B′) with entries in a commutative ring R are automorphically conjugate over the matrix algebra M ₂(R), under an additional assumption that both pairs generate M ₂(R) as an R-algebra.     

The Frobenius exponent of Cartier subalgebras

Let R be a standard graded finitely generated algebra over an F-finite field of prime characteristic, localized at its maximal homogeneous ideal. In this note, we prove that the Frobenius complexity of R is finite. Moreover, we extend this result to Cartier subalgebras of R.     

On the action of derivations on nilpotent ideals of associative algebras

Let I be a nilpotent ideal of an associative algebra A over a field F and let D be a derivation of A. We prove that the ideal I + D(I) is nilpotent if char F = 0 or the nilpotency index I is less than char F = p in the case of the positive characteristic of the field F. In particular, the sum N(A) of all nilpotent ideals of the algebra A is a characteristic ideal if char F = 0 or N(A) is a nilpotent ideal of index < p = char F.     

PRINCIPAL IDEAL DOMAINS AND EUCLIDEAN DOMAINS HAVING 1 AS THE ONLY UNIT

We consider a question raised by Mowaffaq Hajja about the structure of a principal ideal domain R having the property that 1 is the only unit of R. We also examine this unit condition for the case where R is a Euclidean domain. We prove that a finitely generated Euclidean domain having 1 as its only unit is isomorphic to the field with two elements F ₂ or to the polynomial ring F ₂[X]. On the other hand, we establish existence of finitely generated principal ideal domains R such that 1 is the only unit of R and R is not isomorphic to F ₂ or to F ₂[X]. We also construct principal ideal domains R of infinite transcendence degree over F ₂ with the property that 1 is the only unit of R.

     

Efficient Generation of Prime Ideals in Polynomial Rings up to Radical

We call an ideal I of a ring R radically perfect if among all ideals whose radical is equal to the radical of I, the one with the least number of generators has this number of generators equal to the height of I. Let R be a ring and R[X] be the polynomial ring over R. We prove that if R is a strong S-domain of finite Krull dimension and if each nonzero element of R is contained in finitely many maximal ideals of R, then each maximal ideal of R[X] of maximal height is the J _max-radical of an ideal generated by two elements. We also show that if R is a Prüfer domain of finite Krull dimension with coprimely packed set of maximal ideals, then for each maximal ideal M of R, the prime ideal MR[X] of R[X] is radically perfect if and only if R is of dimension one and each maximal ideal of R is the radical of a principal ideal. We then prove that the above conditions on the Prüfer domain R also imply that a power of each finitely generated maximal ideal of R is principal. This result naturally raises the question whether the same conditions on R imply that the Picard group of R is torsion, and we prove this to be so when either R is an almost Dedekind domain or a Prüfer domain with an extra condition imposed on it.     

Derivations with algebraic values of bounded degree

Let Rbe a prime algebra over a field .F, d a nonzero derivation of Rand ρ a nonzero right ideal of R. Suppose that for every x∈ ρ,d(x) is algebraic over Fof bounded degree. Then Ris a primitive ring with a minimal right ideal eR, where e=e² ∈Rand eReis a finite-dimensional central division algebra, except when dis an inner derivation induced by an element a in the two-sided Martindale quotient ring of Rsuch that aρp = 0. An analogous result is also proved for the Lie ideal case.     

On the Schur multiplier of augmented algebras

Letk be a field, andA a finitely generatedk-algebra, with augmentation. Suppose there is a presentation ofA 0→I→R→A→0 whereR is a finitely generated freek-algebra andI is non-zero. IfA is infinite dimensional overk, Lewin proved thatR/I ² is not finitely presented. A stronger statement would be that the ‘Schur multiplier’ ofR/I ² is not finite dimensional. In the case thatA is an augmented domain, we prove this stronger statement, and some related statements.     

Resolutions of monomial almost complete intersections and generalizations

Let S=K[x₁,…,x_n] be a polynomial ring over a field kand let / be a monomial ideal of S. The main result of this paper is an explicit minimal resolution of kover R= S/Iwhen / is a monomial almost complete intersection ideal of S. We also compute an upper bound on the multigraded resolution of k over a generalization of monomial almost complete intersection ring.     

Generic tropical initial ideals of Cohen-Macaulay algebras

We study the generic tropical initial ideals of a positively graded Cohen-Macaulay algebra R over an algebraically closed field k. Building on work of Römer and Schmitz, we give a formula for each initial ideal, and we express the associated quasivaluations in terms of certain I-adic filtrations. As a corollary, we show that in the case that R is a domain, every initial ideal coming from the codimension 1 skeleton of the tropical variety is prime, so “generic presentations of Cohen-Macaulay domains are well-poised in codimension 1.”     

Kronecker Function Rings of Transcendental Field Extensions

We consider the ring Kr(F/D), where D is a subring of a field F, that is the intersection of the trivial extensions to F(X) of the valuation rings of the Zariski–Riemann space consisting of all valuation rings of the extension F/D and investigate the ideal structure of Kr(F/D) in the case where D is an affine algebra over a subfield K of F and the extension F/K has countably infinite transcendence degree, by using the topological structure of the Zariski–Riemann space. We show that for any pair of nonnegative integers d and h, there are infinitely many prime ideals of dimension d and height h that are minimal over any proper nonzero finitely generated ideal of Kr(F/D).     

Nilpotent orbits over ground fields of good characteristic

Let X be an F-rational nilpotent element in the Lie algebra of a connected and reductive group G defined over the ground field F. Suppose that the Lie algebra has a non-degenerate invariant bilinear form. We show that the unipotent radical of the centralizer of X is F-split. This property has several consequences. When F is complete with respect to a discrete valuation with either finite or algebraically closed residue field, we deduce a uniform proof that G(F) has finitely many nilpotent orbits in (F). When the residue field is finite, we obtain a proof that nilpotent orbital integrals converge. Under some further (fairly mild) assumptions on G, we prove convergence for arbitrary orbital integrals on the Lie algebra and on the group. The convergence of orbital integrals in the case where F has characteristic 0 was obtained by Deligne and Ranga Rao (1972).     

Identities of PI-Algebras Graded by a Finite Abelian Group

We consider associative PI-algebras over an algebraically closed field of zero characteristic graded by a finite abelian group G. It is proved that in this case the ideal of graded identities of a G-graded finitely generated PI-algebra coincides with the ideal of graded identities of some finite dimensional G-graded algebra. This implies that the ideal of G-graded identities of any (not necessary finitely generated) G-graded PI-algebra coincides with the ideal of G-graded identities of the Grassmann envelope of a finite dimensional (G × ?₂)-graded algebra, and is finitely generated as GT-ideal. Similar results take place for ideals of identities with automorphisms.     

Power-Central Elements in Tensor Products of Symbol Algebras

Let A be a central simple algebra over a field F. Let k₁,…, k_r be cyclic extensions of F such that k₁ ?_F… ?_Fk_r is a field. We investigate conditions under which A is a tensor product of symbol algebras where each field k_i lies in a symbol F-algebra factor of the same degree as k_i over F. As an application, we give an example of an indecomposable algebra of degree 8 and exponent 2 over a field of 2-cohomological dimension 4.     

Constants of Derivation and Differential Ideals

ABSTRACT

Let R be a differential domain finitely generated over a differential field F of characteristic 0. Let C be the subfield of differential constants of F. This paper investigates conditions on differential ideals of R that are necessary or sufficient to guarantee that C is also the set of constants of differentiation of the quotient field, E, of R. In particular, when C is algebraically closed and R has a finite number of height one differential prime ideals, there are no new constants in E. An example where F is infinitely generated over C shows the converse is false. If F is finitely generated over C and R is a polynomial ring over F, sufficient conditions on F are given so that no new constants in E does imply only finitely many height one prime differential ideals in R. In particular, F can be Q¯(T) where T is a finite transcendence set.

     

The distribution of prime divisors in finitely generated domains

In this paper we show that each divisor class of a finitely generated k-algebra R (k a field) contains a prime divisor if k is Hilbertian or if dim R≥ 2. On the way we also obtain partial results for domains finitely generated over an one-dimensional noetherian domain. Received: 27 June 1998     

The Rees Valuations of Complete Ideals in a Regular Local Ring

Let I be a complete m-primary ideal of a regular local ring (R, m) of dimension d ≥ 2. In the case of dimension two, the beautiful theory developed by Zariski implies that I factors uniquely as a product of powers of simple complete ideals and each of the simple complete factors of I has a unique Rees valuation. In the higher dimensional case, a simple complete ideal of R often has more than one Rees valuation, and a complete m-primary ideal I may have finitely many or infinitely many base points. For the ideals having finitely many base points Lipman proves a unique factorization involving special *-simple complete ideals and possibly negative exponents of the factors. Let T be an infinitely near point to R with dim R = dim T and R/m = T/m _T . We prove that the special *-simple complete ideal P _RT has a unique Rees valuation if and only if either dim R = 2 or there is no change of direction in the unique finite sequence of local quadratic transformations from R to T. We also examine conditions for a complete ideal to be projectively full.