Let (T,M) be a complete local domain containing the integers. Let p1 ⊆ p2 ⊆ · · · ⊆ pn be a chain of nonmaximal prime ideals of T such that Tpn is a regular local ring. We construct a chain of excellent local domains An ⊆ An−1 ⊆ · · · ⊆ A1 such that for each 1 ≤ i ≤ n, the completion of Ai is T , the generic formal fiber of Ai is local with maximal ideal pi, and if I is a nonzero ideal of Ai th...

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...

Maximal length sequences (m-sequences) are also known as pseudo random sequences or pseudo noise sequences for closely following Golomb’s popular randomness properties: (P1) balance, (P2) run, and (P3) ideal autocorrelation. Apart from these, there also exist certain other less known properties of such sequences all of which are discussed in this tutorial paper. Comprehensive proofs to each of ...

Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. An $R$-module $M$ is called a multiplication module if for every submodule $N$ of $M$ there exists an ideal $I$ of $R$ such that $N = IM$. It is shown that over a Noetherian domain $R$ with dim$(R)leq 1$, multiplication modules are precisely cyclic or isomorphic to an invertible ideal of $R$. Moreover, we give a charac...

Abstract Let R be a discrete valuation domain with field of fractions Q and maximal ideal generated by $\pi $ . $\Lambda an -order such that $Q\Lambda is separable -algebra. Maranda showed there exists $k\in \mathbb {N}$ for all -lattices L M , if $L/L\pi ^k\simeq M/M\pi ^k$ then $L\simeq M$ Moreover, complete indecomposable -lattice, also indecomposable. We extend Maranda’s theorem to the clas...

It is well known that the set of all ideals(2) of a ring forms a complete modular lattice with respect to set inclusion. The same is true of the set of all right ideals. Our purpose in this paper is to consider the consequences of imposing certain additional restrictions on these ideal lattices. In particular, we discuss the case in which one or both of these lattices is complemented, and the c...

Introduction. There has arisen in recent years a substantial body of work on “multiplier ideals” in commutative rings (see [La]). Multiplier ideals are integrally closed ideals with properties that lend themselves to highly interesting applications. One is tempted then to ask just how special multiplier ideals are among integrally closed ideals in general. In this note we show that in a two-dim...