Let R be a countable, principal ideal domain which is not a field and A be a countable R-algebra which is free as an R-module. Then we will construct an aleph_1-free R-module G of rank aleph_1 with endomorphism algebra End_RG=A . Clearly the result does not hold for fields. Recall that an R-module is aleph_1-free if all its countable submodules are free, a condition closely related to Pontryagin's theorem. This result has many consequences, depending on the algebra A in use. For instance, if we choose A=R, then clearly G is an indecomposable `almost free' module. The existence of such modules was unknown for rings with only finitely many primes like R=Z_{(p)}, the integers localized at some prime p. The result complements a classical realization theorem of Corner's showing that any such algebra is an endomorphism algebra of some torsion-free, reduced R-module G of countable rank. Its proof is based on new combinatorial-algebraic techniques related with what we call rigid tree-elements coming from a module generated over a forest of trees.
The dimensions of the homomorphism spaces to indecomposable modules over the four subspace algebra
这是论文“On the Multiplicity Problem and the Isomorphism Problem for the Four Subspace Algebra”的附录代数通讯，40：6（2012），2005-2036（DOI：10.1080/00927872.2011.570830）。我们在这里给出了命题3.3的完整证明，描述了四个子空间代数Λ上的不可分解模的同态空间的维数公式。读者也可以在这里找到不可分解的Λ-模的完整描述。
Categorification of Verma modules and indecomposable projective modules in the category $\mathcal I_{\mathfrak{g}}(\mathfrak{sl}_2)$ for $\mathfrak{sl}_2$
利用T. T的一个张量积分解定理，对$\mathfrak{sl}_2$的Verma和不可分解投射模进行范畴$\mathcal I_{\mathfrak{g}}(\mathfrak{sl}_2)$的分类. J.Enright和J.Chuang和R. Rouquier，A. Licata和A. Savage（Hecke代数、有限一般线性群和Heisenberg分类）和M. Khovanov（Heisenberg Algebra and a Graphical Calculus）.
Rings of Invariant Module Type and Automorphism-Invariant Modules
一个模被称为自同构不变的，如果它在其内射壳的任何自同构下是不变的。在[Algebras for which every indecomable right modules is invariant in its injective envelope，Pacific J. Math.，第31卷，第3期（1969），655-658] Dickson和Fuller证明了如果$R$是一个域$\mathbb F$上的有限维代数，该域有两个以上的元素，那么一个不可分解的自同构不变右$R$ -模必须是拟内射的。在本文中，我们表明，这一结果不成立，如果$\mathbb F$是一个域的两个元素。Dickson和Fuller进一步证明了，如果$R$是一个域$\mathbb F$上的有限维代数，且该域有两个以上的元素，那么$R$是右不变模类型当且仅当每个不可分解的右$R$ -模都是自同构不变的。本文将Dickson和Fuller的结果推广到任意的右Artin环.一个环$R$称为右自同构不变型（简称RAI型），如果每一个环生成的不可分解右$R$ -模都是自同构不变的.本文完全刻划了不可分解的RAI型右Artin环。
