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...

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...

The lattice conditional independence (LCI) model N(C)<) is defined to be the set of all normal distributions N(O, E) on IR I such that for every pair L, M E 1<, XL and xM are conditionally independent given xLnM (ct , [AP] (1993a)). Here C)< is a ring of subsets (hence a distributive lattice) of the finite index set I such that 0, IE ".K, while for K E 1<, XK is the coordinate projection of x E...

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)(...

Sometimes the derivation is denoted by a→ a′. A ring equipped with a derivation is called a differential ring. The notions, differential subring, differential ring extension and homomorphism between differential rings, are clear. If (R, ∂) is a differential ring then the set of constants is by definition {r ∈ R : ∂(r) = 0}, and is easily seen to be a (differential) subring. ∂ means the k-fold i...

2. Since A[b] is a subring of B, it is an integral domain. Thus if bz = 0 and b = 0, then z = 0. 3. Any linear transformation on a finite-dimensional vector space is injective iff it is surjective. Thus if b ∈ B and b = 0, there is an element c ∈ A[b] ⊆ B such that bc = 1. Therefore B is a field. 4. Since P is the preimage of Q under the inclusion map of A into B, P is a prime ideal. The map a ...

Let T be a torus and B a compact T−manifold. Goresky, Kottwitz, and MacPherson show in [GKM] that if B is (what was subsequently called) a GKM manifold, then there exists a simple combinatorial description of the equivariant cohomology ring H∗ T (B) as a subring of H∗ T (B ). In this paper we prove an analogue of this result for T−equivariant fiber bundles: we show that if M is a T−manifold and...

and we know that this ring is isomorphic to Z or an order O in an imaginary quadratic field K; in fact, the ring on the right is equal to Z or O (viewed as a subring of C).1 To simplify the discussion, we shall treat the isomorphism in (1) as an equality and view elements of End(EL) as elements of Z or O. How might we construct an elliptic curve with endomorphism ring O? An obvious way is to us...

Let X be a d-dimensional projective arithmetic variety and L an invertible sheaf on X . We fix a C∞-hermitian metric | · | of L. In Arakelov geometry, we frequently ask whether H(X,L) has a free basis consisting of strictly small sections. However, we know few about this problem in general. For example, Zhang [7] proved that H(X,nL) possesses a free basis as above for n ≫ 1 if the following con...