Let K be a complete field with a discrete valuation v, ring of integers OK , and maximal ideal (πK). Let k := OK/(πK) be the residue field, assumed to be separably closed of characteristic p ≥ 0. LetX/K be a smooth proper geometrically irreducible curve of genus g ≥ 1. Let X /OK denote a regular model of X/K. Let Xk = ∑v i=1 riCi be its special fiber, where Ci/k is an irreducible component of X...

In work spread over several decades, Dynkin ([4, 3]), Seitz ([10, 11]), and Testerman ([16]) classified the maximal closed connected subgroups of simple algebraic groups. Their analyses for the classical group cases were based primarily on a striking result: If G is a simple algebraic group and φ : G SL V is a tensor indecomposable irreducible rational representation, then with specified except...

We study an interesting class of Banach function algebras of innitely dierentiable functions onperfect, compact plane sets. These algebras were introduced by Honary and Mahyar in 1999, calledLipschitz algebras of innitely dierentiable functions and denoted by Lip(X;M; ), where X is aperfect, compact plane set, M = fMng1n=0 is a sequence of positive numbers such that M0 = 1 and(m+n)!Mm+n ( m!Mm)...

let $r$ be a commutative ring with identity. let $g(r)$ denote the maximal graph associated to $r$, i.e., $g(r)$ is a graph with vertices as the elements of $r$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $r$ containing both. let $gamma(r)$ denote the restriction of $g(r)$ to non-unit elements of $r$. in this paper we study the various graphi...

By first describing the fixed maximal ideals of RL and those of its bounded part, R∗L, we show that every fixed maximal ideal of the bigger ring contracts to a fixed maximal ideal of the smaller ring, and every fixed maximal ideal of the smaller ring extends to a fixed maximal ideal of the bigger ring. However, the only instance where every maximal ideal of R∗L extends to a maximal ideal of RL ...

Solution: The ring F [x] of polynomials with coefficients in a field F is a P.I.D. Each prime ideal is generated by a monic, irreducible polynomial. Assume there are only a finite number of prime ideals generated by the polynomials f1, . . . , fn and let f(x) = 1+f1(x) · · · fn(x). No fi divides f , hence f is also irreducible. This contradicts the assumption that all the prime ideals were gene...

A finitely generated $R$-module is said to be a module of type ($F_r$) if its $(r-1)$-th Fitting ideal is the zero ideal and its $r$-th Fitting ideal is a regular ideal. Let $R$ be a commutative ring and $N$ be a submodule of  $R^n$ which is generated by columns of  a matrix $A=(a_{ij})$ with $a_{ij}in R$ for all $1leq ileq n$, $jin Lambda$, where $Lambda $ is a (possibly infinite) index set.  ...

Let G be a linear algebraic group defined over Q, and assume that G(R) is compact and meets every connected component of G(C). Let Q̂ := Ẑ ⊗ Q be the ring of finite adèles. Every arithmetic subgroup Γ of G(Q) is finite, and is obtained by choosing an open, compact subgroup K of G(Q̂) and defining Γ = K ∩G(Q) in G(Q̂). We note that G(Q) is discrete and co-compact in G(Q̂). In this paper, we consider...

1. Maximal subgroups. I will assume in this section that the finite simple groups can be classified and even that their irreducible representations in characteristicp can someday be determined. What I wish to demonstrate is that this will carry us a long way toward the determination of their maximal subgroups. It should not come as a total surprise that such a problem is tractible, since the co...