论文标题
近似环的本地紧凑型模型
Locally compact models for approximate rings
论文作者
论文摘要
通过戒指的大约子来，我们的意思是一个添加性的对称子集$ x $，因此$ x \ cdot x \ cup（x +x）$涵盖了有限的许多添加剂翻译为$ x $。我们证明，每个戒指的$ x $每个大约近似$ x $都有一个局部紧凑的型号，即戒指同构$ f \ colon \ colon \ langle x \ rangle x \ rangle \ to s $ to s $ to s $ to s $ to s $ s $ s $，因此$ f [x] $在$ s $中是$ 0 $ $ $ u $ $ u $ u $ u $ u $ u $ f^ - 4x+x \ cdot 4x $（其中$ 4X：= x+x+x+x $）。此$ S $作为环$ \ langle x \ rangle $的商的商品，该$通过其类型可定义的环连接组件在充分饱和的模型中解释。 以上定理可以看作是近似子环的一般结构结果：每个近似的子$ x $都可以通过本地紧凑的模型$ f \ colon \ langle x \ langle x \ rangle \作为$ 0 $ 0 $ $ 0 $ in $ s $ s $ s $ n $ 0的s $恢复为添加剂的可分解性。它还导致更精确的结构甚至分类结果。例如，我们推断出，正面特征的每个[可定义]近似子$ x $都可以与$ 4X + x \ cdot 4x $中的[可定义]子包含在附加性上。 This implies that for any given $K,L \in \mathbb{N}$ there exists $C(K,L)$ such that every $K$-approximate subring $X$ (i.e. $K$ additive translates of $X$ cover $X \cdot X \cup (X+X)$) of a ring of positive characteristic $\leq L$ is additively $C(K,L)$-commensurable with a $ 4X + x \ cdot 4x $中包含的子包含。 We also deduce a classification of finite approximate subrings of rings without zero divisors: for every $K \in \mathbb{N}$ there exists $N(K) \in \mathbb{N}$ such that for every finite $K$-approximate subring $X$ of a ring without zero divisors either $|X| <n（k）$或$ 4X + x \ cdot 4x $是一个子来$ k^{11} $ - 与$ x $相称。
By an approximate subring of a ring we mean an additively symmetric subset $X$ such that $X\cdot X \cup (X +X)$ is covered by finitely many additive translates of $X$. We prove that each approximate subring $X$ of a ring has a locally compact model, i.e. a ring homomorphism $f \colon \langle X \rangle \to S$ for some locally compact ring $S$ such that $f[X]$ is relatively compact in $S$ and there is a neighborhood $U$ of $0$ in $S$ with $f^{-1}[U] \subseteq 4X + X \cdot 4X$ (where $4X:=X+X+X+X$). This $S$ is obtained as the quotient of the ring $\langle X \rangle$ interpreted in a sufficiently saturated model by its type-definable ring connected component. The above theorem can be seen as a general structural result about approximate subrings: every approximate subring $X$ can be recovered up to additive commensurability as the preimage by a locally compact model $f \colon \langle X \rangle \to S$ of any relatively compact neighborhood of $0$ in $S$. It also leads to more precise structural or even classification results. For example, we deduce that every [definable] approximate subring $X$ of a ring of positive characteristic is additively commensurable with a [definable] subring contained in $4X + X \cdot 4X$. This implies that for any given $K,L \in \mathbb{N}$ there exists $C(K,L)$ such that every $K$-approximate subring $X$ (i.e. $K$ additive translates of $X$ cover $X \cdot X \cup (X+X)$) of a ring of positive characteristic $\leq L$ is additively $C(K,L)$-commensurable with a subring contained in $4X + X \cdot 4X$. We also deduce a classification of finite approximate subrings of rings without zero divisors: for every $K \in \mathbb{N}$ there exists $N(K) \in \mathbb{N}$ such that for every finite $K$-approximate subring $X$ of a ring without zero divisors either $|X| <N(K)$ or $4X + X \cdot 4X$ is a subring which is additively $K^{11}$-commensurable with $X$.
