Narrow Results

Let $R$ be a ∗-ring. Then $R$ is called a ∗-symmetric ring if for any $a,b,c\in R,abc=0$ implies $ac{b}^{\ast }=0.$ Obviously, ∗-symmetric ring is certainly symmetric ring, while ∗-ring and symmetric ring does not equal ∗-symmetric ring. Even though reduced rings are symmetric, reduced rings need not be ∗-symmetric ring. In this paper, we further add some conditions for symmetric ring to be ∗-symmetric and reduced. As an application, we discover if $R$ be a ∗-symmetric ring, then $R^{\#}=R^{EP}=R^{+}=R^{\mathrm{\text{reg}}}.$ Also, we expand the definition of ∗-symmetric ring to different combinations of $a,b,c$. The properties of ∗-symmetric ring play a role in generalized inverses of elements in rings.

It is well known that the lattice of subspaces of a vector space over a field is modular. We investigate under which conditions this lattice is orthocomplemented with respect to the orthogonality operation. Using this operation, we define closed subspaces of a vector space and study the lattice of these subspaces. In particular, we investigate when this lattice is modular or orthocomplemented. Finally, we introduce splitting subspaces as special closed subspaces and we prove that the poset of splitting subspaces and the poset of projections are isomorphic orthomodular posets. The vector spaces under consideration are of arbitrary dimension and over arbitrary fields.

Let $R$ be a ring with involution ∗. An element $a\in R$ is called ∗-strongly regular if there exists a projection $p$ of $R$ such that $p\in {\mathrm{\text{comm}}}^2(a)$, $ap=0$ and $a+p$ is invertible, and $R$ is said to be ∗-strongly regular if every element of $R$ is ∗-strongly regular. We discuss the relations among strongly regular rings, ∗-strongly regular rings, regular rings and ∗-regular rings. Also, we show that an element $a$ of a ∗-ring $R$ is ∗-strongly regular if and only if $a$ is EP. Hence we finally give some characterizations of EP elements.

Right C-restriction semigroups are the common generalizations of right C-rpp semigroups and right C-lpp semigroups. The structure of such a semigroup is established in terms of the $\mathrm{\Delta }$-product of a right regular band and a C-restriction semigroup. As its applications, we give the structures of right C-rpp semigroups and right C-lpp semigroups. Also, we give the dual wreath product structure of right C-restriction semigroups. These results extend the related results in (Semigroup Forum 68 (2004) 280–292) and (Algebra Colloquium 14 (2007) 285–294).