In this paper, we present some applications of quaternions and octonions. We present the real matrix representations for complex octonions and some of their properties which can be used in computations where these elements are involved. Moreover, we give a set of invertible elements in split quaternion algebras and in split octonion algebras.

All rings in this paper are commutative with unity; we will deal mainly with integral domains. Let R be a ring with total quotient ring K. A fractional ideal I of R is invertible if II−1 = R; equivalently, I is a projective module of rank 1 (see, e.g., [Eis95, Section 11.3]). Here, I−1 = (R : I) = {x ∈ K |xI ⊆ R}. Moreover, a projective R-module of rank 1 is isomorphic to an invertible ideal. (...

Let M and N be complex unital Jordan-Banach algebras, let M−1 N−1 denote the sets of invertible elements in N, respectively. Suppose that M⊆M−1 N⊆N−1 are clopen subsets N−1, respectively, which closed for powers, inverses products form Ua(b). In this paper we prove each surjective isometry Δ:M→N there exists a real-linear T0:M→N an element u0 McCrimmon radical such Δ(a)=T0(a)+u0 all a∈M. Assumi...

Invertible almost invariant sets The rst problem is in our treatment of almost invariant sets which are invertible (see De nition 2.12). Before discussing the details, we need to brie y recall the construction in chapter 3. We have a nitely generated group G with nitely generated subgroups H1; : : : ; Hn and, for 1 i n, we have a nontrivial Hi{almost invariant subset Xi of G. Recall that E deno...

In this paper, by virtue of the properties generalized inverses elements in rings with involution, we construct related equations. By discussing solutions these equations, invertible are characterized.

Let M(R) denote the measure algebra on the additive group of the reals. R. G. Douglas recently pointed out to us the importance of the following question in the study of Wiener-Hopf integral equations: if fxÇ:M(R) is invertible, then under what conditions does jit = exp(j>) for some vGM(R)? The relevance of the above question in integral equations stems from the fact that if JJLÇÏM(R) is invert...

Let a be a non-invertible transformation of a finite set and let G be a group of permutations on that same set. Then 〈G, a 〉 \G is a subsemigroup, consisting of all non-invertible transformations, in the semigroup generated by G and a. Likewise, the conjugates ag = g−1ag of a by elements g ∈ G generate a semigroup denoted 〈ag | g ∈ G〉. We classify the finite permutation groups G on a finite set...