Example 1.1. A field homomorphism K → F is a character by restricting it to the nonzero elements of K (that is, using G = K×) and ignoring the additive aspect of a field homomorphism. In particular, when L/K is a field extension any element of Aut(L/K) is a field homomorphism L→ L and therefore is a character of L× with values in L×. Example 1.2. For any α ∈ F×, the map Z→ F× by k 7→ αk is a ch...

It is not difficult to see that every group homomorphism from Z to R extends to a homomorphism from R to R. We discuss other examples of discrete subgroups Γ of connected Lie groups G, such that the homomorphisms defined on Γ can (“virtually”) be extended to homomorphisms defined on all of G. For the case where G is solvable, we give a simple proof that Γ has this property if it is Zariski dens...

Let X be a smooth projective variety defined over an algebraically closed field, and let L be an ample line bundle over X . We prove that for any smooth hypersurface D on X in the complete linear system |L|, the inclusion map D →֒ X induces an isomorphism of fundamental group schemes, provided d is sufficiently large and dimX ≥ 3. If dimX = 2, and d is sufficiently large, then the induced homomo...

Let S be a distributive {∨, 0 }-semilattice. In a previous paper, the second author proved the following result: Suppose that S is a lattice. Let K be a lattice, let φ : Conc K → S be a {∨, 0 }-homomorphism. Then φ is, up to isomorphism, of the form Conc f , for a lattice L and a lattice homomorphism f : K → L. In the statement above, Conc K denotes as usual the {∨, 0 }-semilattice of all finit...

This study investigates the effect of different link beam lengths in the Reinforced Concrete (RC) frame retrofitted with the Linked Column Frame (LCF) system. It also investigates the ratio of the link beam length (e) to the span length of the RC frame (L) from 0 to 1.5 for the 9 models of the RC frame retrofitted by the LCF system has been investigated. In addition, it studies the formation of...

a topoframe, denoted by $l_{ tau}$,  is a pair $(l, tau)$ consisting of a frame $l$ and a subframe $ tau $ all of whose elements are complementary elements in$l$. in this paper, we define and study the notions of a$tau $-real-continuous function on a frame $l$ and the set of realcontinuous functions $mathcal{r}l_tau $ as an $f$-ring.we show that $mathcal{r}l_{ tau}$is actually a generalization ...

In this paper, we study some of the properties of (Q, L) -fuzzy normal ideal of a ring and prove some results on these. 2000 AMS SUBJECT CLASSIFICATION: 03F55, 08A72, 20N25.

Let $nin mathbb{N}$. An additive map $h:Ato B$ between algebras $A$ and $B$ is called $n$-Jordan homomorphism if $h(a^n)=(h(a))^n$ for all $ain A$. We show that every $n$-Jordan homomorphism between commutative Banach algebras is a $n$-ring homomorphism when $n < 8$. For these cases, every involutive $n$-Jordan homomorphism between commutative C-algebras is norm continuous.

let $mathcal {a} $ and $mathcal {b} $ be c$^*$-algebras. assume that $mathcal {a}$ is of real rank zero and unital with unit $i$ and $k>0$ is a real number. it is shown that if $phi:mathcal{a} tomathcal{b}$ is an additive map preserving $|cdot|^k$ for all normal elements; that is, $phi(|a|^k)=|phi(a)|^k $ for all normal elements $ainmathcal a$, $phi(i)$ is a projection, and there exists a posit...