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Let M and N be modules over a commutative ring R with N Noetherian. We define the injective capacity of M with respect to N over R to be the supremum of the values t for which N ⊕t embeds into M . In a dual fashion, we deem the number of cogenerators of N with respect to M over R to be the infimum of the numbers t for which N embeds into M ⊕t . We demonstrate that the global injective capacity is the infimum of its local analogues and that the global number of cogenerators is the supremum of the corresponding local invariants. We also prove enhanced versions of these statements and consider the graded case.

A result of Monsky states that the Hilbert-Kunz function of a one-dimensional local ring of prime characteristic has a term φ that is eventually periodic. For example, in the case of a power series ring in one variable over a prime-characteristic field, φ is the zero function and is therefore immediately periodic with period 1. In additional examples produced by Kunz and Monsky, φ is immediately periodic with period 2. We show that, for every positive integer π, there exists a ring for which φ is immediately periodic with period π.Let (R, m) be a one-dimensional commutative Noetherian local ring of prime characteristic p. In this note, we study the Hilbert-Kunz function HK m,R , which sends every nonnegative integer e to the length of R/m [p e ] over R, where m [p e ] := (r p e : r ∈ m).The general form of HK m,R is known: Monsky has proved that there exist a positive integer e HK (m, R), called the Hilbert-Kunz multiplicity of R, and an eventually periodic function) for every nonnegative integer e [4, Theorem 3.10].Also, there are algorithms for specialized settings: Kreuzer has covered the case in which R = ŜM , where S is a standard graded algebra over a field and M is the maximal homogeneous ideal of S [2, Algorithm 5.4]. Upadhyay has obtained a formula valid in the event that S is a polynomial ring over a field modulo a binomial with zero constant term (and R = ŜM with M the maximal homogeneous ideal of S once again) [5, Theorem 26].Despite these advances, explicit values of HK m,R are still scarce. Moreover, when values are available, φ m,R does not differ much from one example to the next: For instance, in the six computations achieved by Kreuzer [2, Examples 6.1-6.6], the function φ m,R is eventually constant in four cases, and in the remaining two examples, which are attributable to Kunz [3, Example 4.6b] and Monsky [4, page 46], the function φ m,R is immediately periodic with period 2.In response to the lack of diversity in concrete examples of φ m,R , we show here that, for every positive integer π, there exists a ring (R, m) for which φ m,R is immediately periodic with period π. To accomplish our goal, we first provide an infinite family of rings whose Hilbert-Kunz functions we can compute explicitly. To illustrate the content of our result, we recover Monsky's example with period 2 and give new examples with periods 3 and 4. We then appeal to a classical observation of Dirichlet to prove our main theorem.

The Module Cancellation Problem solicits hypotheses that, when imposed on modules K, L, and M over a ring S, afford the implicationIn a well-known paper on basic element theory from 1973, Eisenbud and Evans lament the "great scarcity of strong results" in module cancellation research, expressing the wish that, "under some general hypothesis" on finitely generated modules over a commutative Noetherian ring, cancellation could be demonstrated. Singling out cancellation theorems by Bass and Dress that feature "large" projective modules, Eisenbud and Evans contend further that, although "[s]ome criteria of 'largeness' is certainly necessary in general [. . . ,] the need for projectivity is not clear." In this paper, we prove that cancellation holds if K, L, and M are finitely generated modules over a commutative Noetherian ring S such that K ⊕(1+dim(S/p)) p is a direct summand of M p over S p for every prime ideal p of S. We also weaken projectivity conditions in the cancellation theorems of Bass and Dress and a newer theorem by De Stefani-Polstra-Yao; in fact, we obtain a statement that unifies all three of these theorems while obviating a projectivity constraint in each one. To illustrate the scope of our work, we construct a cancellation example that simultaneously eludes the three theorems just mentioned as well as many other observations from the module cancellation literature.