Abstract Let ℛ be a semiprime ring with unity e and ϕ , φ automorphisms of ℛ. In this paper it is shown that if satisfies 2

Tits has defined Kac–Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley–Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac–Moody groups whose Dynkin diagrams are hyperbolic and simply laced. Our presentations are analogues of the Curtis–Tits presentation of the finite groups of Lie type. Whe...

We present an algorithm that returns a proper factor of a polynomial Φ(x) over the p-adic integers Zp (if Φ(x) is reducible over Qp) or returns a power basis of the ring of integers of Qp[x]/Φ(x)Qp[x] (if Φ(x) is irreducible over Qp). Our algorithm is based on the Round Four maximal order algorithm. Experimental results show that the new algorithm is considerably faster than the Round Four algo...

Let $$\underline{S}$$ be an arbitrary subset of $$R^n$$ where R is a domain with the field fractions $$\mathbb {K}$$ . Denote ring polynomials in n variables over by {K}[\underline{x}] $$ The integer-valued , denoted Int $$(\underline{S},R)$$ defined as set which maps to R. In this article, we study irreducibility for first time case when Unique Factorization Domain. We also show that our resul...

Let D be a division ring and let m,n be integers ≥ 2. Let Mm×n(D) be the space of m × n matrices. In the fundamental theorem of the geometry of rectangular matrices all bijective mappings φ of Mm×n(D) are determined such that both φ and φ−1 preserve adjacency. We show that if a bijective map φ of Mm×n(D) preserves the adjacency then also φ −1 preserves the adjacency. Thus the supposition that φ...

There are many instances of the principle that if A is an algebra of functions on X, then every ring homomorphism A → C is given by evaluation at a particular x ∈ X. Examples are the nullstellensatz in algebraic geometry and the result of Gelfand when X is a compact Hausdorff space. In these cases one can regard X as being included in Hom(A, C) as the set of those f : A → C which satisfy the se...

The Briançon-Skoda theorem appears in many variations in recent literature. The common denominator is that the theorem gives a sufficient condition that implies a membership φ ∈ a, where a is an ideal of some ring R. In the analytic interpretation R is the local ring of an analytic space Z, and the condition is that |φ| ≤ C|a| holds on the space Z. The theorem thus relates the rate of vanishing...