Proposition 1. Let R be a ring and M an R-module. Then EndR(M), the set of R-linear maps from M to M , is a subring of End(M). Proof. Recall from 3.1.6 that the ring (End(M), +, ◦) is defined by (α + β)(m) = α(m) + β(m), (α ◦ β)(m) = α(β(m)). We must verify the four subring conditions from 3.2.2. (i) The additive identity of End(M) is z : M → M, m → 0M . Note that for any m,n ∈ M z(m + n) = 0M ...

Let A be a finitary algebra over a finite field k, and A-mod the category of finite dimensional left A-modules. Let H(A) be the corresponding Hall algebra, and for a positive integer r let Dr(A) be the subspace of H(A) which has a basis consisting of isomorphism classes of modules in A-mod with at least r+1 indecomposable direct summands. If A is hereditary of type An, then Dr(A) is known to be...

Vague language is observed in elements that make the meaning unclear. In this pragma-linguistic paper, authors aim to review different proposed theories and definitions of vague were suggested middle past century. Through discourse analysis usage, justify politeness communication, gender, culture, press releases, education. Authors found there dissatisfaction among speakers with describing term...

In 1978, Gareth Evans published a short and somewhat cryptic
 article purporting to establish that there are no vague objects. This
 paper is commentary on this. Prima facie, the claim there
 objects clearly false. Mt Everest, for example, has no
 precise boundaries. And if this so, must be something wrong
 with Evans' argument. paper, I discuss what is, giving a
 ...

In SCIS 2017, Choi and Kim introduced the new linearly homomorphic ring signature scheme (CK17 scheme) based on the hardness of SIS problem, which overcomes the limitation of Boneh and Freeman’s scheme to implement homomorphic signatures to the real world scenario under multiple signers setting for a message. They replace the original sampling algorithm SamplePre() by Gentry et al. with Wang an...

An ideal K of R is a subset that is both a left ideal and a right ideal of R. For emphasis, we sometimes call it a two-sided ideal but the reader should understand that unless qualified, the word ideal will always refer to a two-sided ideal. The zero ideal (0) and the whole ring R are examples of two-sided ideals in any ring R. A (left)(right) ideal I such that I 6= R is called a proper (left)(...