On representability of *-regular and regular involutive rings in endomorphism rings of vector spaces

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The origins of von Neumann-regular and *-regular rings trace back to the work of John von Neumann and F. J. Murray in the 1930s, linking operator theory, ring theory, and lattice theory. This historical context suggests that von Neumann may have drawn inspiration from both operator and lattice theories when introducing the concept of a *-regular ring. The positivity requirement of the involution reflects a generalization of operator algebra involutions, while *-regular rings correspond to a significant class of lattices associated with operator algebras. This thesis posits that the *-regular ring framework effectively describes regular rings of operators, particularly when considering vector spaces over involutive skew fields with a scalar product. Key findings include that every *-regular ring is representable in this context, and each variety of *-regular rings is generated by its simple Artinian members. Additionally, the thesis explores regular involutive rings and their representability. Jacobson's work on rings without involution established that representability as subrings of endomorphism rings is linked to primitivity. For involutive rings, the concepts of *-primitivity and representations via bi-vector spaces are introduced, alongside an examination of primitive rings with involution to develop a suitable non-degenerated form on vector spaces. This research builds on the work of Herrmann, Micol, and Niemann, culminat