The notions of irreducible ideals of the endomorphism ring on the category of rings and the category of modules (original) (raw)

Ring of Endomorphisms and Modules over a Ring

Formalized Mathematics

Summary We formalize in the Mizar system [3], [4] some basic properties on left module over a ring such as constructing a module via a ring of endomorphism of an abelian group and the set of all homomorphisms of modules form a module [1] along with Ch. 2 set. 1 of [2]. The formalized items are shown in the below list with notations: Mab for an Abelian group with a suffix “ ab ” and M without a suffix is used for left modules over a ring. 1. The endomorphism ring of an abelian group denoted by End(Mab ). 2. A pair of an Abelian group Mab and a ring homomorphism R → ρ R\mathop \to \limits^\rho End (Mab ) determines a left R-module, formalized as a function AbGrLMod(Mab, ρ) in the article. 3. The set of all functions from M to N form R-module and denoted by Func_Mod R (M, N). 4. The set R-module homomorphisms of M to N, denoted by Hom R (M, N), forms R-module. 5. A formal proof of Hom R (¯R, M) ≅M is given, where the ¯R denotes the regular representation of R, i.e. we regard R itself a...

On commutative endomorphism rings

Pacific Journal of Mathematics, 1970

This note deals with a finitely generated faithful module E over a commutative semi-prime noetherian ring R, with commutative endomorphism ring Hom J2 (E r , E) = Ω(E). It is shown that E is identifiable to an ideal of R whenever Ω(E) lacks nilpotent elements; a class of examples with Ω(E) commutative but not semi-prime is discussed.

Endomrphism rings via minimal morphisms

2020

We prove that if u:K → M is a left minimal extension, then there exists an isomorphism between two subrings, End_R^M(K) and End_R^K(M) of End_R(K) and End_R(M) respectively, modulo their Jacobson radicals. This isomorphism is used to deduce properties of the endomorphism ring of K from those of the endomorphism ring of M in certain situations such us when K is invariant under endomorphisms of M, or when K is invariant under automorphisms of M.

Modules which are reduced over their endomorphism rings

2015

Let RRR be an arbitrary ring with identity and MMM a right RRR-module with S=S=S= End$_R(M)$. The module MMM is called {\it reduced} if for any minMm\in MminM and finSf\in SfinS, fm=0fm=0fm=0 implies fMcapSm=0f M\cap Sm=0fMcapSm=0. In this paper, we investigate properties of reduced modules and rigid modules.

On the arithmetic of the endomorphisms ring

2012

For a prime number p, Bergman (1974) established that End(Z p × Z p 2) is a semilocal ring with p 5 elements that cannot be embedded in matrices over any commutative ring. We identify the elements of End(Z p × Z p 2) with elements in a new set, denoted by E p , of matrices of size 2 × 2, whose elements in the rst row belong to Z p and the elements in the second row belong to Z p 2 ; also, using the arithmetic in Z p and Z p 2 , we introduce the arithmetic in that ring and prove that the ring End(Z p × Z p 2) is isomorphic to the ring E p. Finally, we present a Die-Hellman key interchange protocol using some polynomial functions over E p dened by polynomial in Z[X]. x + y = (x + y) mod m and x • y = (xy) mod m, for all x, y ∈ Z m. Let us assume from now on that p is a prime number and consider the rings Z p and Z p 2. Clearly, we can also assume that Z p ⊆ Z p 2 , even though Z p is not a subring of Z p 2. Then, it follows that notation is utmost important to prevent errors like the following. Suppose that p = 5, then Z 5 = {0, 1, 2, 3, 4} and Z 5 2 = {0, 1, 2, 3,. .. , 23, 24}. Note that 2, 4 ∈ Z 5 and 2 + 4 = 1 ∈ Z 5 ; but 2, 4 ∈ Z 5 2 equally. However when 2, 4 ∈ Z 5 2 , 2 + 4 = 6 ∈ Z 5 2. Obviously, 1 = 6 in Z 5 2. Such error can be easily avoidable if we write, when necessary, x mod p and x mod p 2 to refer the element x when x ∈ Z p and x ∈ Z p 2 , respectively. In this light, the above example could be rewritten as (2 mod 5)+(4 mod 5) = 1 mod 5, whereas (2 mod 5 2) + (4 mod 5 2) = 6 mod 5 2. 2 The ring End(Z p × Z p 2) Consider the additive group Z p × Z p 2 of order p 3 , where the addition is dened componentwise, and the set End(Z p × Z p 2) of endomorphisms of such additive group. It is well known that End(Z p × Z p 2) is a noncommutative unitary ring with the usual addition and composition of endomorphisms, that are dened, for f, g ∈ End(Z p × Z p 2), as (f + g)(x, y) = f (x, y) + g(x, y) and (f • g)(x, y) = f (g(x, y)). The additive and multiplicative identities O and I are dened, obviously, by O(x, y) = (0, 0) and I(x, y) = (x, y) respectively. The additive identity is also called the null endomorphism. Te next result not only determines the cardinality of the ring End(Z p × Z p 2), but also introduces the primary property of such a ring: it cannot be embedded in matrices over any commutative ring. Theorem 1 (Theorem 3 of [2]) If p is a prime number, then the ring of endomorphisms End(Z p × Z p 2) has p 5 elements and is semilocal, but cannot be embedded in matrices over any commutative ring.