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Ring of integers
About: Ring of integers is a research topic. Over the lifetime, 1856 publications have been published within this topic receiving 15882 citations.
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TL;DR: In this paper, the authors studied the linear inhomogeneous first order differential equation by ρ + f(x) = y in the ring of formal power series with integer coefficients and showed that the Euler series can be interpreted as the fundamental solution to the equation under consideration.
Abstract: We study the linear inhomogeneous first order differential equation by′ + f(x) = y in the ring of formal power series with integer coefficients. Using the p-adic topology on the ring of integers, we construct a counterpart of the Hurwitz product of the Euler series and an arbitrary formal power series with integer coefficients. It is shown that the Euler series can be interpreted as the fundamental solution to the equation under consideration. Bibliography: 8 titles.
5 citations
TL;DR: Left zero divisor graph of the ring R, where R is the set of all 3 × 3 matrices over the ring of integers modulo 2 is constructed and algorithm to compute vertices and edges of this graph is presented.
Abstract: In this article left zero divisor graph of the ring R, where R is the set of all 3 × 3 matrices over the ring of integers modulo 2 is constructed We present algorithm to compute the vertices and e
5 citations
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TL;DR: In this paper, a triangulated monoidal Karoubi closed category with Grothendieck ring is constructed, naturally isomorphic to the ring of integers localized at two.
Abstract: We construct a triangulated monoidal Karoubi closed category with the Grothendieck ring, naturally isomorphic to the ring of integers localized at two.
5 citations
TL;DR: In this paper, the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over the ring of integers in a number field, and the same result holds also for spinor spinor integral regular quadrastic forms over ℤ.
Abstract: Let 𝔬 be the ring of integers in a number field. An integral quadratic form over 𝔬 is called regular if it represents all integers in 𝔬 that are represented by its genus. In [13,14] Watson proved that there are only finitely many inequivalent positive definite primitive integral regular ternary quadratic forms over ℤ. In this paper, we generalize Watson's result to totally positive regular ternary quadratic forms over ${\mathbb Z}[\frac{1 + \sqrt{5}}{2}]$. We also show that the same finiteness result holds for totally positive definite spinor regular ternary quadratic forms over ${\mathbb Z}[\frac{1 + \sqrt{5}}{2}]$, and thus extends the corresponding finiteness results for spinor regular quadratic forms over ℤ obtained in [1,3].
5 citations
TL;DR: In this article, the unique minimal ambiguous ideal U of an algebraic number field and a Galois extension of it is denoted by M (K, K′) in terms of the ramification invariants of ramified primes in L F.
5 citations