graph theory - Is there a purely set-theoretic expression of the Euler characteristic?

noncommutative algebra – What is the rife standing of the Kaplansky zero-divisor surmise for group rings? retort

Hello pricey customer to our community We will proffer you an answer to this query noncommutative algebra – What is the rife standing of the Kaplansky zero-divisor surmise for group rings? ,and the retort will breathe typical by way of documented data sources, We welcome you and proffer you recent questions and solutions, Many customer are questioning in regards to the retort to this query.

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:

  1. 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?
  2. 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.

we are going to proffer you the answer to noncommutative algebra – What is the rife standing of the Kaplansky zero-divisor surmise for group rings? query through our community which brings all of the solutions from a number of reliable sources.

Add comment