Mathematics

Given any finite ring, we'll construct a partition of it, where each block corresponds to one idempotent. The partition is so simple that it works for any finite power-associative algebra (in particular for any finite associative, Jordan, or Cayley-Dickson algebra.) We prove in particular that idempotents and regular elements can always be lifted (over homomorphisms) for finite rings.

Motivated by a polynomial identity of certain iterated integrals, first observed in [CGM20] in the setting of lattice paths, we prove an intriguing combinatorial identity in the shuffle algebra. It has a close connection to de Bruijn's formula when interpreted in the framework of signatures of paths.

We give a new purely algebraic proof of the Baker-Campbell-Hausdorff theorem, which states that the homogeneous components of the formal expansion of log( e A e B ) are Lie polynomials. Our proof is based on a recurrence formula for these components and a lemma that states that if under certain conditions a commutator of a non-commuting variable and a given polynomial is a Lie polynomial, then the given polynomial itself is a Lie polynomial.

Let A and B be commutative algebras and n⩾2 an integer. Then each n− Jordan homomorphism h:A→B is an n− homomorphism.

A thin Lie algebras is a Lie algebra L , graded over the positive integers, with its first homogeneous component L 1 of dimension two and generating L , and such that each nonzero ideal of L lies between consecutive terms of its lower central series. All its homogeneous components have dimension one or two, and the two-dimensional components are called diamonds. We prove that if the next diamond past L 1 of an infinite-dimensional thin Lie algebra L is L k , with k>5 , then [Lyy]=0 for some nonzero element y of L 1 .

In this paper we prove that if R is a proper alternative ring whose additive group has no 3 -torsion and whose non-zero commutators are not zero-divisors, then R has no zero-divisors. It follows from a theorem of Bruck and Kleinfeld that if, in addition, the characteristic of R is not 2, then the central quotient of R is an octonion division algebra over some field. We include other characterizations of octonion division algebras and we also deal with the case where (R,+) has 3 -torsion.

In this paper, we provide some solvability conditions in terms of ranks for the existence of a general solution to a system of k Sylvester-type quaternion matrix equations with 3k+1 variables A i X i + Y i B i + C i Z i D i + F i Z i+1 G i = E i , i= 1,k ¯ ¯ ¯ ¯ ¯ ¯ ¯ . As applications of this system, we present rank equalities as the necessary and sufficient conditions for the existence of a general solution to some systems of quaternion matrix equations A i X i +( A i X i ) ϕ + C i Z i ( C i ) ϕ + F i Z i+1 ( F i ) ϕ = E i , i= 1,k ¯ ¯ ¯ ¯ ¯ ¯ ¯ .

For a finite group G and U:=U(ZG) , the group of units of the integral group ring of G , we study the implications of the structure of G on the abelianization U/ U ′ of U . We pose questions on the connections between the exponent of G/ G ′ and the exponent of U/ U ′ as well as between the ranks of the torsion-free parts of Z(U) , the center of U , and U/ U ′ . We show that the units originating from known generic constructions of units in ZG are well-behaved under the projection from U to U/ U ′ and that our questions have a positive answer for many examples. We then exhibit an explicit example which shows that the general statement on the torsion-free part does not hold, which also answers questions from [BJJ + 18].

Given an integral commutative residuated lattice L=(L,\vee,\wedge), its full twist-product (L^2,\sqcup,\sqcap) can be endowed with two binary operations \odot and \Rightarrow introduced formerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. Wille such that it becomes a commutative residuated lattice. For every a in L we define a certain subset P_a(L) of L^2. We characterize when P_a(L) is a sublattice of the full twist-product (L^2,\sqcup,\sqcap). In this case P_a(L) together with some natural antitone involution ' becomes a pseudo-Kleene lattice. If L is distributive then (P_a(L),\sqcup,\sqcap,') becomes a Kleene lattice. We present sufficient conditions for P_a(L) being a subalgebra of (L^2,\sqcup,\sqcap,\odot,\Rightarrow) and thus for \odot and \Rightarrow being a pair of adjoint operations on P_a(L). Finally, we introduce another pair \odot and \Rightarrow of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law and we investigate when P_a(L) is closed under these new operations \odot and \Rightarrow.

In 2014, Wolfgang Rump showed that there exists a correspondence between left nilpotent right R-braces and pre-Lie algebras. This correspondence, established using a geometric approach related to flat affine manifolds and affine torsors, works locally. In this paper we explain Rump's correspondence using only algebraic formulae. An algebraic interpretation of the correspondence works for fields of sufficiently large prime characteristic as well as for fields of characteristic zero.