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noncommutative algebra – What is the rife standing of the Kaplansky zero-divisor surmise for group rings?
Let $Okay$ breathe a bailiwick and $G$ a bunch. The so known as zero-divisor surmise for group rings asserts that the group ring $Okay[G]$ is a province if and provided that $G$ is a torsion-free group.
A pair of wonderful sources for this drawback that offers some historic overview are:
Passman, Donald S. The algebraic construction of group rings. absolute and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], recent York-London-Sydney, 1977.
Passman, Donald S. Group rings, crossed merchandise and Galois idea. CBMS Regional Conference succession in Mathematics, 64. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1986.
The surmise has been confirmed affirmative, when $G$ belongs to particular courses of teams. I attempted to write down down a number of the historical past:
- Ordered teams (A.I. Malcev 1948 and B.H. Neumann 1949)
- Supersolvable teams (E. Formanek 1973)
- Polycyclic-by-finite teams (Okay.A. Brown 1976, D.R. Farkas & R.L. Snider 1976)
- unique product teams (J.M. Cohen, 1974)
Here are my questions:
- Was Irving Kaplansky the primary one to situation this surmise? Can somebody present me with a reference to a paper or bespeak that claims this?
- Since the publications of Passman’s expository level to (above) in 1986, has there been any main developments on the issue? Are there any recent courses of teams that may relent a constructive retort to the surmise? Can somebody help me to increase my record above?
The zero-divisor surmise (let’s denote it by “(Z)”) is said to the next two conjectures:
(I): If $G$ is torsion-free, then $Okay[G]$ has no non-trifling idempotents.
(U): If $G$ is torsion-free, then $Okay[G]$ has no non-trifling models.
Now, if $G$ is torsion-free, then one can array that:
(U) $Rightarrow$ (Z) $Rightarrow$ (I).
Has there been any developments, since 1986, to any partial solutions on surmise (U)? Passman claims that “this is not plane known for supersolvable groups”. Is this quiet the illustration?
I need to level out that this publish is said to a different historic MO-post.
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