Atomistic simulations are employed to investigate chemical short-range ordering in two body-centered cubic refractory multi-principal element alloys, HfMoNbTaTi and HfNbTaTiZr, its influence on their ideal simple shear strengths. Both the alias affine strengths analyzed {110} {112} planes opposing 111 directions. In both quinary local of NbNb, TaTa, HfNb, HfTa, NbTa is preferred as annealing te...

We study the transfer homomorphism in modular invariant theory paying particular attention to the image of the transfer which is a proper non-zero ideal in the ring of invariants. We prove that, for a p-group over Fp whose ring of invariants is a polynomial algebra, the image of the transfer is a principal ideal. We compute the image of the transfer for SLn(Fq) and GLn(Fq) showing that both ide...

The structure of order ideals in the Bruhat order for the symmetric group is elucidated via permutation patterns. A method for determining nonisomorphic principal order ideals is described and applied for small lengths. The permutations with boolean principal order ideals are characterized. These form an order ideal which is a simplicial poset, and its rank generating function is computed. More...

The classification of rings of algebraic integers which are Euclidean (not necessarily for the norm function) is a major unsolved problem. Assuming the Generalized Riemann Hypothesis, Weinberger [7] showed in 1973 that for algebraic number fields containing infinitely many units the ring of integers R is a Euclidean domain if and only if it is a principal ideal domain. Since there are principal...

Let E(p) denote the Eisenstein ideal in the Hecke algebra T(p) of the Drinfeld modular curve X0(p) parameterizing Drinfeld modules of rank two over Fq[T ] of general characteristic with Hecke level p-structure, where p⊳ Fq[T ] is a nonzero prime ideal. We prove that the characteristic p of the field Fq does not divide the order of the quotient T(p)/E(p) and the Eisenstein ideal E(p) is locally ...

The factorization of ζo(s) is the main issue. After giving a definition of this zeta function, we will see that the factorization is equivalent to understanding the behavior of rational primes in the extension ring Z[ω] of Z: do they stay prime, or do they factor as products of primes in Z[ω]? A complication is that the rings Z[ω] are rarely principal ideal domains. To delay contemplation of th...

2 Integral Domains 12 2.1 Factorization in Integral Domains . . . . . . . . . . . . . . . . 12 2.2 Euclidean Domains . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Principal Ideal Domains . . . . . . . . . . . . . . . . . . . . . 16 2.4 Fermat’s Two Squares Theorem . . . . . . . . . . . . . . . . . 17 2.5 Maximal Ideals and Prime Ideals . . . . . . . . . . . . . . . . 20 2.6 Unique Fact...

We introduce the basic notions of ideal theory in rings. This includes left and right ideals, (finitely) generated ideals and some operations on ideals such as the addition of ideals and the radical of an ideal. In addition we introduce linear combinations to formalize the well-known characterization of generated ideals. Principal ideal domains and Noetherian rings are defined. The latter devel...