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The left spectrum, the Levitzki radical, and noncommutative schemes
1Department of Mathematics, Harvard University, Cambridge, MA 02138, USA.
Summary
This study introduces a noncommutative approach to algebraic geometry, assigning schemes to associative rings. The left spectrum of a ring is central to this noncommutative geometric framework.
Area of Science:
- Algebraic Geometry
- Noncommutative Geometry
- Ring Theory
Background:
- Traditional algebraic geometry studies geometric objects using commutative rings.
- Extending these concepts to noncommutative settings presents unique challenges and opportunities.
Purpose of the Study:
- To present the fundamental concepts of a noncommutative version of affine, quasi-affine, and projective algebraic geometry.
- To establish a framework for noncommutative geometric studies.
Main Methods:
- Assigning a quasi-affine (or affine) left scheme to any associative ring with unity.
- Utilizing the concept of the left spectrum of a ring as a key element.
Main Results:
- A foundational exposition of noncommutative algebraic geometry is provided.
- The left spectrum is established as a crucial tool for constructing noncommutative schemes.
Conclusions:
- The developed framework offers a noncommutative perspective on geometric structures.
- This work lays the groundwork for further exploration in noncommutative algebraic geometry.