The sum-product phenomenon in arbitrary rings
نویسنده
چکیده
The sum-product phenomenon predicts that a finite set A in a ring R should have either a large sumset A + A or large product set A ·A unless it is in some sense “close” to a finite subring of R. This phenomenon has been analysed intensively for various specific rings, notably the reals R and cyclic groups Z/qZ. In this paper we consider the problem in arbitrary rings R, which need not be commutative or contain a multiplicative identity. We obtain rigorous formulations of the sum-product phenomenon in such rings in the case when A encounters few zero-divisors of R. As applications we recover (and generalise) several sum-product theorems already in the literature.
منابع مشابه
The unit sum number of Baer rings
In this paper we prove that each element of any regular Baer ring is a sum of two units if no factor ring of R is isomorphic to Z_2 and we characterize regular Baer rings with unit sum numbers $omega$ and $infty$. Then as an application, we discuss the unit sum number of some classes of group rings.
متن کاملRings in which elements are the sum of an idempotent and a regular element
Let R be an associative ring with unity. An element a in R is said to be r-clean if a = e+r, where e is an idempotent and r is a regular (von Neumann) element in R. If every element of R is r-clean, then R is called an r-clean ring. In this paper, we prove that the concepts of clean ring and r-clean ring are equivalent for abelian rings. Further we prove that if 0 and 1 are the only idempotents...
متن کاملThe sum-annihilating essential ideal graph of a commutative ring
Let $R$ be a commutative ring with identity. An ideal $I$ of a ring $R$is called an annihilating ideal if there exists $rin Rsetminus {0}$ such that $Ir=(0)$ and an ideal $I$ of$R$ is called an essential ideal if $I$ has non-zero intersectionwith every other non-zero ideal of $R$. Thesum-annihilating essential ideal graph of $R$, denoted by $mathcal{AE}_R$, isa graph whose vertex set is the set...
متن کاملBEST APPROXIMATION IN QUASI TENSOR PRODUCT SPACE AND DIRECT SUM OF LATTICE NORMED SPACES
We study the theory of best approximation in tensor product and the direct sum of some lattice normed spacesX_{i}. We introduce quasi tensor product space anddiscuss about the relation between tensor product space and thisnew space which we denote it by X boxtimesY. We investigate best approximation in direct sum of lattice normed spaces by elements which are not necessarily downwardor upward a...
متن کاملOn zero-divisor graphs of quotient rings and complemented zero-divisor graphs
For an arbitrary ring $R$, the zero-divisor graph of $R$, denoted by $Gamma (R)$, is an undirected simple graph that its vertices are all nonzero zero-divisors of $R$ in which any two vertices $x$ and $y$ are adjacent if and only if either $xy=0$ or $yx=0$. It is well-known that for any commutative ring $R$, $Gamma (R) cong Gamma (T(R))$ where $T(R)$ is the (total) quotient ring of $R$. In this...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Contributions to Discrete Mathematics
دوره 4 شماره
صفحات -
تاریخ انتشار 2009