在该论文中，我们研究幂等元在弱环中的应用。我们主要考察两个已知的结果，Hindman定理的双算子形式以及IP集上的加法和乘法结构定理。第一个结果指对任意自然数集N上的有穷染色，都存在两个子集A和B，使得A中元素的有穷和与B中元素的有穷积都同色。我们尝试将其推广到更一般的代数结构中。我们首先发现该结果在所有弱环上都成立。由于一些常见的结构如((0,1),+,·)并不是弱环，我们提出了一个更一般的代数结构——充分的贫环（定义6.34），并且我们发现该结果在该结构中依然成立。

    第二个结果大致指对任意IP集A和任意序列L，都存在一个L的和子系统L′，使得L′中元素的任意有穷和与有穷积都在A中。我们提出了一个一般概念叫T-巨大集，并且建立了这个概念的一般化结果（定理4.12）：若S为一弱环，T是(βS,+)的紧子半群，则对(S,+)中任意完备T-巨大集A和任意T-序列L，都存在L的和子系统L′使得其所有有穷和与任意次序的有穷积都在A中。该结果对每个完备T-巨大集给出了能体现其组合性质的T-序列，而IP集上的加法和乘法结构定理以及De建立的C集上类似的结构定理也将作为我们的这个一般化结果的特殊情形。进一步的，我们把这个一般化结果应用到另外一个组合概念——类中心集上，其比C集更弱，并且建立了一个类中心集上的新的组合定理（推论4.25）。此外，我们考察了IP集上的加法和乘法结构定理的一个更强的版本——zigzag版本，其考虑l-序列（由l个序列组成）。它是由Goswami提出，但至今尚未解决，不过Goswami发现该结论对Po集成立，Po集是比IP集更强的概念。我们首先严格加强了他的结果，然后我们还考虑了IP集的情况，并得到定理5.13：对任意IP集A和任意兼容的l-序列L，都存在L的对角和子系统L′，使得L′中元素的任意zigzag有穷和与任意次序的zigzag有穷积都在A中。再者，我们发现当给定的序列是充分时IP集上的加法和乘法结构定理在充分的贫环中依然成立。

    另外，我们也考察了不可数同色集下的Hindman型性质。Fernández-Bretón证明了Hindman定理只是一种可数现象。之后，他和Carlucci考虑了一种特殊的积叫邻近有穷积，并且建立了邻近的Hindman定理。我们考虑这种乘积以及另外一种类似的乘积——连结有穷积。我们得到了一个关于连结有穷积的组合结果（定理7.16）。在特定的大基数假设下，我们也建立了一个组合结果，它可以看成是Rado路径分解定理在不可数情况下的推广。

In this thesis, we study the applications of idempotents to weak rings. We mainly inves- tigate two known results, the two operator version of Hindman theorem and the additive and multiplicative structure theorem for IP sets. The first result says for every finite coloring of natural number set N, there exist two subsets A and B such that the colors of finite sums of elements of A and finite products of elements of B are identical. We attempt to generalize this result to more general algebraic structures. we firstly discover that it holds in all weak rings. Since some common structures like ((0, 1), +, ·) are not weak rings, we come up with a more general algebraic structure - adequate poor rings(Definition 6.34), and we find that the result is still correct in this structure.

    The second result roughly says for any IP set A and any sequence L, there exists a sum subsystem L′ of L such that all finite sums and finite products of elements of L′ are in A. We come up with a general notion called T -large sets, and establish a general result of this notion (Theorem 4.12): if S is a weak ring, T is a compact subsemigroup of (βS, +), then for any completely T -large set A and any T -sequence L in (S, +), there exists a sum subsystem L′ of L such that all finite sums and finite products in any order of elements of L′ are in A. This result gives every completely T -large set corresponding T -sequences to show its combined property. While the additive and multiplicative structure theorem for IP sets as well as De’s result, who established a similar structure theorem for C sets, are as specific cases of our general result. Moreover, we apply this general result to another combinatorial notion - quasi- central sets, which is weaker than C sets, and establish a new combined structure theorem  for quasi-central  sets (Corollary 4.25).  In addition, we investigate    a stronger version of the additive and multiplicative structure theorem for IP sets - zigzag version, which considers l-sequences (consists of l-many sequences). It is asked by Goswami, but it has not been solved yet, however, Goswami found that the conclusion holds for Po sets, this is a notion stronger than IP sets. We firstly strengthen his result strictly, then we consider the case of IP sets, and obtain Theorem 5.13 that for any IP set A and any l-sequence L which need to be compatible, there exists a diagonal sum subsystem L′ of L such that all zigzag finite sums and zigzag finite products in any order of elements of L′ are in A. Furthermore, we also find that the additive and multiplicative structure theorem for IP sets remains true in adequate poor rings when the given sequence is adequate.

    Besides, we also investigate Hindman-type properties for uncountable monochromatic sets. Fernández-Bretón proved that Hindman theorem is only a countable phenomenon, afterwards, he and Carlucci considered a special kind of product called adjacent finite products and then established Adjacent Hindman Theorem. We consider this product and another similar one - syndetic finite products, and obtain a combinatorial result of syndetic finite products(Theorem 7.16). Under certain large cardinal assumptions, we also establish a combined result, which can be viewed as a generalization of Rado’s path decomposition theorem in uncountable situation.