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In this paper, we study expanding phenomena in the setting of matrix rings. More precisely, we will prove that If $A$ is a set of $M_2(mathbb{F}_q)$ and $|A|gg q^{7/2}$, then we have [|A(A+A)|, ~|A+AA|gg q^4.] If $A$ is a set of $SL_2(mathbb{F}_q)$ and $|A|gg q^{5/2}$, then we have [|A(A+A)|, ~|A+AA|gg q^4.] We also obtain similar results for the cases of $A(B+C)$ and $A+BC$, where $A, B, C$ are sets in $M_2(mathbb{F}_q)$.
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In this paper, we study the expanding phenomena in the setting of higher dimensional matrix rings. More precisely, we obtain a sum-product estimate for large subsets and show that x+yz, x(y+z) are moderate expanders over the matrix ring, and xy + z + t is strong expander over the matrix rings. These results generalize recent results of Y.D. Karabulut, D. Koh, T. Pham, C-Y. Shen, and the second listed author.
Let $(R, mathfrak{m})$ be a complete discrete valuation ring with the finite residue field $R/mathfrak{m} = mathbb{F}_{q}$. Given a monic polynomial $P(t) in R[t]$ whose reduction modulo $mathfrak{m}$ gives an irreducible polynomial $bar{P}(t) in mathbb{F}_{q}[t]$, we initiate the investigation of the distribution of $mathrm{coker}(P(A))$, where $A in mathrm{Mat}_{n}(R)$ is randomly chosen with respect to the Haar probability measure on the additive group $mathrm{Mat}_{n}(R)$ of $n times n$ $R$-matrices. One of our main results generalizes two results of Friedman and Washington. Our other results are related to the distribution of the $bar{P}$-part of a random matrix $bar{A} in mathrm{Mat}_{n}(mathbb{F}_{q})$ with respect to the uniform distribution, and one of them generalizes a result of Fulman. We heuristically relate our results to a celebrated conjecture of Cohen and Lenstra, which predicts that given an odd prime $p$, any finite abelian $p$-group (i.e., $mathbb{Z}_{p}$-module) $H$ occurs as the $p$-part of the class group of a random imaginary quadratic field extension of $mathbb{Q}$ with a probability inversely proportional to $|mathrm{Aut}_{mathbb{Z}}(H)|$. We review three different heuristics for the conjecture of Cohen and Lenstra, and they are all related to special cases of our main conjecture, which we prove as our main theorems. For proofs, we use some concrete combinatorial connections between $mathrm{Mat}_{n}(R)$ and $mathrm{Mat}_{n}(mathbb{F}_{q})$ to translate our problems about a Haar-random matrix in $mathrm{Mat}_{n}(R)$ into problems about a random matrix in $mathrm{Mat}_{n}(mathbb{F}_{q})$ with respect to the uniform distribution.
Let $R$ be a commutative local ring. It is proved that $R$ is Henselian if and only if each $R$-algebra which is a direct limit of module finite $R$-algebras is strongly clean. So, the matrix ring $mathbb{M}_n(R)$ is strongly clean for each integer $n>0$ if $R$ is Henselian and we show that the converse holds if either the residue class field of $R$ is algebraically closed or $R$ is an integrally closed domain or $R$ is a valuation ring. It is also shown that each $R$-algebra which is locally a direct limit of module-finite algebras, is strongly clean if $R$ is a $pi$-regular commutative ring.
The commuting graph of a group G, denoted by Gamma(G), is the simple undirected graph whose vertices are the non-central elements of G and two distinct vertices are adjacent if and only if they commute. Let Z_m be the commutative ring of equivalence classes of integers modulo m. In this paper we investigate the connectivity and diameters of the commuting graphs of GL(n,Z_m) to contribute to the conjecture that there is a universal upper bound on diam(Gamma(G)) for any finite group G when Gamma(G) is connected. For any composite m, it is shown that Gamma(GL(n,Z_m)) and Gamma(M(n,Z_m)) are connected and diam(Gamma(GL(n,Z_m))) = diam(Gamma(M(n,Z_m))) = 3. For m a prime, the instances of connectedness and absolute bounds on the diameters of Gamma(GL(n,Z_m)) and Gamma(M(n,Z_m)) when they are connected are concluded from previous results.
The Furstenberg-Sarkozy theorem asserts that the difference set $E-E$ of a subset $E subset mathbb{N}$ with positive upper density intersects the image set of any polynomial $P in mathbb{Z}[n]$ for which $P(0)=0$. Furstenbergs approach relies on a correspondence principle and a polynomial version of the Poincare recurrence theorem, which is derived from the ergodic-theoretic result that for any measure-preserving system $(X,mathcal{B},mu,T)$ and set $A in mathcal{B}$ with $mu(A) > 0$, one has $c(A):= lim_{N to infty} frac{1}{N} sum_{n=1}^N mu(A cap T^{-P(n)}A) > 0.$ The limit $c(A)$ will have its optimal value of $mu(A)^2$ when $T$ is totally ergodic. Motivated by the possibility of new combinatorial applications, we define the notion of asymptotic total ergodicity in the setting of modular rings $mathbb{Z}/Nmathbb{Z}$. We show that a sequence of modular rings $mathbb{Z}/N_mmathbb{Z}$, $m in mathbb{N},$ is asymptotically totally ergodic if and only if $mathrm{lpf}(N_m)$, the least prime factor of $N_m$, grows to infinity. From this fact, we derive some combinatorial consequences, for example the following. Fix $delta in (0,1]$ and a (not necessarily intersective) polynomial $Q in mathbb{Q}[n]$ such that $Q(mathbb{Z}) subseteq mathbb{Z}$, and write $S = { Q(n) : n in mathbb{Z}/Nmathbb{Z}}$. For any integer $N > 1$ with $mathrm{lpf}(N)$ sufficiently large, if $A$ and $B$ are subsets of $mathbb{Z}/Nmathbb{Z}$ such that $|A||B| geq delta N^2$, then $mathbb{Z}/Nmathbb{Z} = A + B + S$.
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