On divisible modules over domains

: In this paper, we study many relationships about Distributive ring and other domain such as Dedekind domain and Noetherian domain. We prove if any divisible module over a ring T, then T is Distributive ring. Also we satisfy if T is invariant ring have multiplication ideal this imply T is a Distributive ring. Finally, we study strongly duo ring and related to Distributive ring.

The concept of physical and fractal dimension I. The projective dimensions

(2) M= Ml GD*.*. Ml with f(M )U=i. Further, we can ask for which decompositions M-M1 = M D Mn can we choose Ml Mi in (2)? We will call an R-module U semiprimary if the ring of multiplications of R on U/rad U is semisimple with minimum condition and U is finitely generated. (All modules considered in this paper will be unitary.) Our Main Theorem 1.2 states that if f:M-M1 C) ...M, onto U = U1 + * + U,, is a homomorphism of R-modules with M projective and each Ui semiprimary, and if, for each i there is some homomorphism of Mi onto Ui, then there is a decomposition (2) for which M M. Note that the sum U1 + *. need not be direct. The main theorem is applied to sharpen a theorem of Steinitz, Chevalley, and others [7, Theorem 22.12] which states that if N is a submodule of a finitely generated, torsion-frr e (hence projective) module M over a Dedekind domain R such that A '/N is a torsion module, then there exist decompositions M = Mi an .. D Mn (Maan ideal of R) and N-EIM, e .. 3 E,Mn (Ei = an ideal). The present contribution to this theorem (Corollary1.10) is that M1, -., Mn -1 can be chosen isomorphic to arbitrary nonzero ideals and the ideals Ei can be chosen subject only to the restriction MIN RIE1 E0 o D R/E, This sharpened version of the theorem turns out to have a more abstract (and slightly more general) formulation (Corollary 1.9): if Ni is a submodule of a projective module Mi (still over a Dedekind domain) such that M1/N1 M2/N2= finitely generated, then M1 M2 if and only if N1 N2; and when the conditions

This study wants to provide a decomposition of the module that was built into a number of special rings, such as the principal ideal domain or the Dedekind domain. The main result of this study is that the decomposition of the finitely generated module is the direct sum of the torsion module and the independent module when the ring is the principal ideal domain. When the ring is the Dedekind domain, the decomposition is the direct sum of the projective module and the torsion module. Abstrak Penelitian ini ingin memberikan dekomposisi dari modul yang dibangun Secara hingga atas beberapa ring khusus, seperti daerah ideal utama atau daerah Dedekind. Hasil utama dari penelitian ini mendapatkan dekomposisi modul yang dibangun secara hingga adalah jumlah langsung dari modul torsi dan modul bebas saat ring adalah daerah ideal utama. Saat ring adalah daerah Dedekind, didapatkan dekomposisi merupakan tambah langsung dari modul projektif dan modul torsi.

Decomposition of integer-valued polynomial algebras

We consider some generalizations of the notion of divisible module, and study when our “divisible” modules coincide with injective modules. Here the base ring $$R$$ is usually a (not-necessarily commutative) integral domain. For instance, we prove that an integral domain $$R$$ is right noetherian and right hereditary if and only if the class of injective right $$R$$ -modules coincides with the class of $$h$$ -divisible right $$R$$ -modules, where “ $$h$$ -divisible” is the non-commutative version of the corresponding notion of $$h$$ -divisible module introduced by Matlis for modules over commutative domains. We then define and study a divisible right $$R$$ -module $$\Delta $$ , which is the non-commutative analogue of the tilting module $$\partial $$ defined by L. Fuchs. The case we are mainly interested in is that of the finitely generated free algebra $$R:=k\langle x_1,x_2,\ldots ,x_n\rangle $$ over a commutative field $$k$$ , where $$x_1,x_2,\ldots ,x_n$$ are non-commuting indeterminates over $$k$$ . For $$n\ge 2$$ , this $$R$$ is a non-noetherian fir, hence, in particular, it is a hereditary integral domain.

Let $T$ be an infinitely generated tilting module of projective dimension at\nmost one over an arbitrary associative ring $A$, and let $B$ be the\nendomorphism ring of $T$. In this paper, we prove that if $T$ is good then\nthere exists a ring $C$, a homological ring epimorphism $B\\ra C$ and a\nrecollement among the (unbounded) derived module categories $\\D{C}$ of $C$,\n$\\D{B}$ of $B$, and $\\D{A}$ of $A$. In particular, the kernel of the total left\nderived functor $T\\otimes_B^{\\mathbb L}-$ is triangle equivalent to the derived\nmodule category $\\D{C}$. Conversely, if the functor $T\\otimes_B^{\\mathbb L}-$\nadmits a fully faithful left adjoint functor, then $T$ is a good tilting\nmodule. We apply our result to tilting modules arising from ring epimorphisms,\nand can then describe the rings $C$ as coproducts of two relevant rings.\nFurther, in case of commutative rings, we can weaken the condition of being\ntilting modules, strengthen the rings $C$ as tensor products of two commutative\nrings, and get similar recollements. Consequently, we can produce examples\n(from commutative algebra and $p$-adic number theory, or Kronecker algebra) to\nshow that two different stratifications of the derived module category of a\nring by derived module categories of rings may have completely different\nderived composition factors (even up to ordering and up to derived\nequivalence),or different lengths. This shows that the Jordan-H\\"older theorem\nfails even for stratifications by derived module categories, and also answers\nnegatively an open problem by Angeleri-H\\"ugel, K\\"onig and Liu.\n

Ideal arithmetic in noetherian Pi rings

Let $H$ be a multiplicatively written monoid. Given $k\in{\bf N}^+$, we denote by $\mathscr U_k$ the set of all $\ell\in{\bf N}^+$ such that $a_1\cdots a_k=b_1\cdots b_\ell$ for some atoms $a_1,\ldots,a_k,b_1,\ldots,b_\ell\in H$. The sets $\mathscr U_k$ are one of the most fundamental invariants studied in the theory of non-unique factorization, and understanding their structure is a basic problem in the field: In particular, it is known that, in many cases of interest, these sets are almost arithmetic progressions with the same difference and bound for all large $k$, namely, $H$ satisfies the Structure Theorem for Unions. The present paper improves the current state of the art on this problem. More precisely, we show that, under mild assumptions on $H$, not only does the Structure Theorem for Unions hold, but there also exists $\mu\in{\bf N}^+$ such that, for every $M\in{\bf N}$, the sequences $$ \bigl((\mathscr U_k-\inf\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1} \quad\text{and}\quad \bigl((\sup\mathscr U_k-\mathscr U_k)\cap[\![0,M]\!]\bigr)_{k\ge 1} $$ are $\mu$-periodic from some point on. The result applies, e.g., to (the multiplicative monoid of) all commutative Krull domains (e.g., Dedekind domains) with finite class group; a variety of weakly Krull commutative domains (including all orders in number fields with finite elasticity); some maximal orders in central simple algebras over global fields; and all numerical monoids. Large parts of the proofs are worked out in a "purely additive model", by inquiring into the properties of what we call a subadditive family, i.e., a collection $\mathscr L$ of subsets of $\bf N$ such that, for all $L_1,L_2\in\mathscr L$, there is $L\in\mathscr L$ with $L_1+L_2\subseteq L$.

The rest of the book is devoted to a study of the equivalence of quadratic forms over the integers of local and global fields. Our first purpose in the present chapter is to state the nature of this problem in modern terminology and in the general setting of an arbitrary Dedekind domain. Our second purpose is to develop some technique in this general situation. The more interesting results must wait until we specialize to the fields of number theory.

One of the aims of the modern representation theory is to solve classification problems for subcategories of modules over a unitary ring R. In this paper, we introduce the concept of 1-absorbing comultiplication modules and classify 1-absorbing comultiplication modules over local Dedekind domains and we study it in detail from the classification problem point of view. The main purpose of this article is to classify all those indecomposable 1-absorbing comultiplication modules with finite-dimensional top over pullback rings of two local Dedekind domains and establish a connection between the 1-absorbing comultiplication modules and the pure-injective modules over such rings. In fact, we extend the definition and results given in [17] to a more general 1-absorbing comultiplication modules case.

The purpose of this paper is to introduce the study of perfect torsion theories on Mod ⁡ R \operatorname {Mod} R and to dualize the concept of divisible module. A torsion theory on Mod ⁡ R \operatorname {Mod} R is called perfect if every torsion module has a projective cover. It is shown for such a theory that the class of torsion modules is closed under projective covers if and only if the class of torsion free modules is closed under factor modules. In addition, it is shown that this condition on a perfect torsion theory is equivalent to its idempotent radical being an epiradical. Codivisible covers of modules are also introduced and we are able to show that any module which has a projective cover has a codivisible cover. Codivisible covers are then characterized in terms of the projective cover of the module and the torsion submodule of the kernel of the minimal epimorphism.

The paper is concerned with properties of an ill-posed problem for the Helmholtz equation when Dirichlet and Neumann conditions are given only on a part ! of the boundary!. We present an equivalent formulation of this problem in terms of a moment problem defined on the part of the boundary where no boundary conditions are imposed. Using a weak definition of the normal derivative, we prove the equivalence between these two problems for an arbitrary Lipschitz domain inR d . Moreover, uniqueness of the solution is proved for the general case when ! is a non-empty open subset of the Lipschitz boundary.

We consider the problem of characterizing the commutative domains such that every module admits a divisible envelope or a [Formula: see text]-cover, where [Formula: see text] is the class of modules of projective dimension at most one. We prove that each one of the two conditions is satisfied exactly for the class of almost perfect domains, that is commutative domains such that all of their proper quotients are perfect rings. Moreover, we show that the class of weak-injective modules over a commutative domain is closed under direct sums if and only if the domain is almost perfect.

Projective modules: Modules and homomorphisms Projective modules Completely reducible modules Wedderbum rings Artinian rings Hereditary rings Dedekind domains Projective dimension Tensor products Local rings Polynomial rings: Skew polynomial rings Grothendieck groups Graded rings and modules Induced modules Syzygy theorem Patching theorem Serre conjecture Big projectives Generic flatness Nullstellensatz Injective modules: Injective modules Injective dimension Essential extensions Maximal ring of quotients Classical ring of quotients Goldie rings Uniform dimension Uniform injective modules Reduced rank Index.