### Properties of Pseudo-Local Rings Dr. Jason Huffman, Georgia College Mathematics Department

#### Abstract

Let $R$ be an associative ring, not necessarily commutative and not necessarily having unity. We say $R$ is a**local**ring if the Jacobson radical of $R$ is a unique maximal left ideal. In parts of the literature, the term

**quasi-local**refers to a ring satisfying the previous definition, but not satisfying the ascending chain condition on left ideals. Such rings play a prominent role in commutative algebra and algebraic geometry and have been studied primarily in the category of rings with identity. Here, we consider the definition above, but in the context of rings without unity. We call $R$ a

**(left) pseudo-local**ring if and only if $R$ has a unique maximal left ideal. If $R$ has unity, this definition coincides with the familiar definition of "local." However, a pseudo-local ring without unity may have properties that differ significantly from those for local rings. For example, a pseudo-local ring may be equal to its Jacobson radical. Here, general properties of pseudo-local rings are explored, and several examples are given to illustrate differences between local and pseudo-local rings.

Keywords and phrases. Ring, Local, Unity, Maximal Left Ideal, Jacoboson Radical

2000 Mathematics Subject Classification: Primary 16L30, 16N20