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Introduction. In a recent paper [2] R. H. Bruck has introduced the concept of right neoring and discussed some properties of these systems. In particular, he has considered analogues of certain properties of the ring of integers. This paper is essentially a commentary on Bruck's paper and we generalize some of his results as follows. The construction of the universal right neoring in [2] is applied to the free monogenic $3-loop in any subvariety $3 of the variety of loops and a complete analogue of Theorem 4.1 of [2] is obtained for any one of these subvarieties. Then, using a result similar to those obtained in [5], it is shown that this construction yields uncountably many right neorings with an identity which generates the additive loop of the right neoring. Conversely, every right neoring with an identity which generates its additive loop can be obtained from a free monogenic $3-loop by the above construction. Each of these right neorings has some properties resembling those of the ring of integers. One possible answer is given to the question raised by Bruck concerning the existence of universal right neorings with free additive loop of arbitrary rank. A brief proof is given, using the results of [4; 5], of the cancellation properties of the monogenic universal right neoring. Finally, we discuss briefly the relationship between right neorings and the logarithmetics of Etherington.

It is shown that the rings under the title admit an order-theoretical characterization as in the commutative case studied by Abian. Introduction. Let R be an associative ring equipped with the binary relation (^) defined by xay if and only if xy = x2 in R. In this paper, it is shown that ( ^ ) is an order relation on R if and only if, R has no nilpotent elements i9*0). Conditions on the binary relation (g) in order that R split into a direct product of division rings are then studied in the light of Abian's result (l, Theorem l). Using similar argumentation and using certain subdirect representation of rings with no nilpotent elements, one obtains a complete similarity with the commutative case (yet, no extra complication in the computa- tions). Conventions. R is an associative ring which is, unless otherwise stated, with no nilpotent elements (other than 0). As a result of (2), R can be embedded into a direct product of skewdomains, R—* YLiei £i (that is to say, rings R, having no one-sided divisors of zero). The former embedding is fixed throughout the paper. It is therefore legiti- mate to identify any element x in R with the family consisting of all its projections (xj.e/. Finally, all definitions in (l) are extended (verbatim) to the present case (of a noncommutative ring R) and are freely used throughout.

It has recently been found that my previous paper “On generalized semi-primary rings and Krull-Remak-Schmidt’s theorem” Jap. Journ. Math. 19 (1949) — referred to as S. K. — contained in its Theorems 19 and 20 some errors. Nevertheless the writer has been able to correct them in suitable forms so that most parts of both theorems hold, even under a weaker assumption, and also subsequent theorems remain valid. These will be, together with some supplementary remarks, shown in the present note.

Recently Cameron and Storvick introduced and studied an operator valued function space integral related to the Feynman integral. The main theorems of their study establish the existence of the function space integral as a weak operator limit of operators defined at the first stage by finite-dimensional integrals. This paper provides a substantial strengthening of their existence theorem giving the function space integrals as strong operator limits rather than as weak operator limits.

The paper ‘A Homological Foundation for Scale Problems in Physics’ (Atkin & Bastin, 1970) is criticised on account of several inconsistencies in the argument. Possible applications of the general ideas used are then discussed in the context of a ‘quantum logic’ type of framework.

In two recent papers of Conner and Floyd ((2)) and ((3)), the additive structure of the SU-cobordism (or bordism) ring was completely determined. The object of this note is to point out that their results can also be used to determine the somewhat complicated multiplicative structure.

The circumstances explained in the footnote on p. 61 of the former paper with this title might well have necessitated a re-writing of the whole. Fortunately it appears that only a few short comments are required. For example it may be noted that the first question in §4 can be answered by counting the degrees of freedom in the two configurations. Eight points of a twisted cubic have freedom 20; four pairs of planes drawn at random through four lines of a regulus have freedom 21; therefore the eight planes of the second paragraph of §4 cannot always lead back to the eight points of §2. This is corroborated by the corresponding numbers in [4], which are 31 and 34.

Everitt has shown [1[, that for α ∊ [0, π/2] the undernoted problem (1.1–2) with an indefinite weight function r can be represented by a selfadjoint operator in a suitable Hilbert space. This result is extended to arbitrary α ∊ [0, π), replacing the Hilbert space in some cases by a Pontrjagin space with index one. The problem is also treated in the Krein space generated by the weight function r.

In the first section of this note we recall the generalized Hodge conjecture (G.H.C.) and its correction as stated by Grothendieck (G.G.H.C.).

It is proved (without the assumption of the continuum hypothesis) that under the condition there exist on the Marcinkiewicz space M(ψ) abnormal functionals which are not localizable.

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A ring R is called, according to [2], a left p.p. ring if any principal left ideal of R is projective. A ring which is left and right p.p. is called a p.p. ring.In this short note we shall give some additional remarks to A. Hattori [2]. In Proposition 1 we shall give a characterization of commutative p.p. rings, and in Proposition 3 we shall give a generalization of Proposition 17 and 18 in [2], which shows also that the modified torsion theory over commutative p.p. rings coincides with the usual torsion theory.

The main goal of this short paper is to prove the existence of rank 2 simple and special Ulrich bundles on a wide class of elliptic surfaces: namely, on regular Weierstrass fibrations \(\pi : S\rightarrow \mathbb {P}^1\). Alongside we also show the existence of rank 2 weakly Ulrich sheaves on arbitrary Weierstrass fibrations \(S\rightarrow C_0\) and we deal with the (non-)existence of rank one Ulrich bundles on them.

(1973). An Addendum to the Paper “A Characterization of the n × n Matrices Over a Finite Field”. The American Mathematical Monthly: Vol. 80, No. 9, pp. 1041-1043.

In this short paper we combine the representability theorem introduced in [17, 18] with the theory of derived formal models introduced in [2] to prove the existence representability of the derived Hilbert space RHilb(X) for a separated k-analytic space X. Such representability results relies on a localization theorem stating that if X is a quasi-compact and quasi-separated formal scheme, then the \infty-category Coh^+(X^rig) of almost perfect complexes over the generic fiber can be realized as a Verdier quotient of the \infty-category Coh^+(X). Along the way, we prove several results concerning the the \infty-categories of formal models for almost perfect modules on derived k-analytic spaces., 28 pages

In this short paper we prove a parametric version of the Harnack inequality for \(\phi\)-Laplacian equations. In this sense, the estimates are optimal and represent an improvement of previous bounds for this kind of operators.