In this note we give examples of a ring that has no maximal ideals. Recall that, by a Zorn’s lemma argument, a ring with identity has a maximal ideal. Therefore, we need to produce examples of rings without identity. To help motivate our examples, let S be a ring without identity. We may embed S in a ring R with identity so that S is an ideal of R. Notably, set R = Z⊕S, as groups, and where mul...

A subset selection Z assigns to each partially ordered set P a certain collection ZP of subsets. The theory of topological and of algebraic (i.e. finitary) closure spaces extends to the general Z-level, by replacing finite or directed sets, respectively, with arbitrary ‘Z-sets’. This leads to a theory of Z-union completeness, Z-arity, Z-soberness etc. Order-theoretical notions such as complete ...

In this paper, we study the internal stabilizability and internal stabilization problems for multidimensional (nD) systems. Within the fractional representation approach, a multidimensional system can be studied by means of matrices with entries in the integral domain of structurally stable rational fractions, namely the ring of rational functions which have no poles in the closed unit polydisc...

Let K[〈X〉] denote the large polynomial ring (in the sense of Halter-Koch [7]) on the set X = {x1, x2, x3, . . . } of indeterminates. For each integer n, there is a truncation homomorphism ρn : K[〈X〉] → K[x1, . . . , xn]. If I is a homogeneous ideal of K[〈X〉], then the N-graded Hilbert series of K[x1,...,xn] ρn(I) can be written as gn(t) (1−t) ; it was shown in [13] that if in addition I is what...

For Banach algebras A in a class which includes all group and function algebras, we show that the family of ideals of A with the same hull is typically quite large, containing ascending and descending chains of arbitrary length through any ideal in the family, and that typically a closed ideal of A whose hull meets the Silov boundary of A cannot be countably generated algebraically. Guided by t...

It is proved that EJ is injective if E is an injective module over a valuation ring R, for each prime ideal J 6= Z. Moreover, if E or Z is flat, then EZ is injective too. It follows that localizations of injective modules over h-local Prüfer domains are injective too. If S is a multiplicative subset of a noetherian ring R, it is well known that SE is injective for each injective R-module E. The...

Let φ : Rm → Rd be a map of free modules over a commutative ring R. Fitting’s Lemma shows that the “Fitting ideal,” the ideal of d × d minors of φ, annihilates the cokernel of φ and is a good approximation to the whole annihilator in a certain sense. In characteristic 0 we define a Fitting ideal in the more general case of a map of graded free modules over a Z/2graded skew-commutative algebra a...

We call (i?) 1 the Lie ring associated with R, and denote it by 9Î. The question of how far the properties of SR determine those of R is of considerable interest, and has been studied extensively for the case when R is an algebra, but little is known of the situation in general. In an earlier paper the author investigated the effect of the nilpotency of 9î upon the structure of R if R contains ...

A generalization of injective modules (noted GI-modules), distinct from p-injective modules, is introduced. Rings whose p-injective modules are GI are characterized. If M is a left GI-module, E = End(AM), then E/J(E) is von Neumann regular, where J(E) is the Jacobson radical of the ring E. A is semisimple Artinian if, and only if, every left A-module is GI. If A is a left p. p., left GI-ring su...

Let R be a ring with center Z, Jacobson radical J , and set N of all nilpotent elements. Call R generalized periodic-like if for all x ∈ R \ (N ∪ J ∪ Z) there exist positive integers m, n of opposite parity for which xm − xn ∈ N ∩ Z. We identify some basic properties of such rings and prove some results on commutativity. Let R be a ring; and let N = N(R), Z = Z(R) and J = J(R) denote respective...