We raise the following general question regarding a ring graded by a group: “If P is a ring-theoretic property, how does one define the graded version P_gr of the property P in a meaningful way?”. Some properties of rings have straightforward and unambiguous generalizations to their graded versions and these generalizations satisfy all the matching properties of the nongraded case. If P is either being unit-regular, having stable range 1 or being directly finite, that is not the case. The first part of the paper addresses this issue. Searching for appropriate generalizations, we consider graded versions of cancellation, internal cancellation, substitution, and module-theoretic direct finiteness. In the second part of the paper, we consider graded matrix and Leavitt path algebras. If K is a trivially graded field and E is a directed graph, the Leavitt path algebra L_K(E) is naturally graded by the ring of integers. If E is a finite graph, we present a property of E which is equivalent with L_K(E) being graded unit-regular. This property critically depends on the lengths of paths to cycles and it further illustrates that graded unit-regularity is quite restrictive in comparison to the alternative generalization of unit-regularity from the first part of the paper.

对于由组进行分级的环，我们提出以下一般性问题：“如果P是环理论性质，那么如何以有意义的方式定义性质P的分级版本P _gr？”。环的某些属性对其分级版本具有简单明了的概括，并且这些概括满足未分级情况的所有匹配属性。如果P要么是单位规则的，具有稳定范围1，要么是直接有限的，事实并非如此。本文的第一部分解决了这个问题。为了寻找适当的概括，我们考虑了抵消，内部抵消，替代和模块理论直接有限性的分级版本。在本文的第二部分中，我们考虑了分级矩阵和Leavitt路径代数。如果K是一个平凡的梯度场，而E是一个有向图，则Leavitt路径代数L _K（E）自然是由整数环分级的。如果E是一个有限图，我们给出E的一个属性，它等于L _K（E）按常规进行分级。该属性严格取决于循环路径的长度，它进一步说明，与本文第一部分对单元规则性的替代概括相比，分级的单元规则性具有很大的局限性。