Surveys in Contemporary Mathematics by Nicholas Young, Yemon Choi

By Nicholas Young, Yemon Choi

Younger scientists in Russia are carrying on with the phenomenal culture of Russian arithmetic of their domestic kingdom, inspite of the post-Soviet diaspora. This assortment, the second one of 2, showcases the hot achievements of younger Russian mathematicians and the powerful examine teams they're linked to. the 1st assortment desirous about geometry and quantity conception; this one concentrates on combinatorial and algebraic geometry and topology. The articles are customarily surveys of the new paintings of the learn teams and comprise a considerable variety of new effects. issues lined comprise algebraic geometry over Lie teams, cohomological elements of toric topology, the Borsuk partition challenge, and embedding and knotting of manifolds in Euclidean areas. The authors are A. E. Guterman, I. V. Kazachkov, A. V. Malyutin, D. V. Osipov, T. E. Panov, A. M. Raigorodskii, A. B. Skopenkov and V. V. Ten

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Algebra, 13:53–70, 2004. In Russian. [9] L. B. Beasley and A. E. Guterman. Rank inequalities over semirings. Journal of Korean Math. , 42(2):223–241, 2005. Rank and determinant functions for matrices over semirings 31 [10] L. B. Beasley, A. E. -B. -Z. Song. Linear preservers of extremes of rank inequalities over semirings: row and column ranks. , 413(2-3):495–509, 2006. [11] L. B. Beasley, A. E. -G. -Z. Song. Linear transformations preserving the Grassmannian over Mn (Z+ ). , 393:39–46, 2004. [12] L.

Let B be an A-domain and Y be an arbitrary subset of B n . Then the closure of Y in the Zariski topology coincides with VB (RadB (Y )). Proof. Clearly the set VB (RadB (Y )) is closed and contains Y . We show that VB (RadB (Y )) is contained in every closed set Z such that Y ⊆ Z. 5, RadB (Y ) ⊇ RadB (Z) and thus VB (RadB (Y )) ⊆ VB (RadB (Z)). 3, every closed set in B n is algebraic over B, hence VB (RadB (Z)) = Z and the statement follows. 6 The category of algebraic sets In this section we introduce the category ASA,B of algebraic sets over an A-Lie algebra B.

An element r ∈ R is called irreducible if r is not invertible in R and for any factorization r = r1 r2 , r1 , r2 ∈ R, either r1 or r2 is invertible in R. 10. A commutative ring R is called a unique factorization domain if the following conditions are satisfied. (i) R has no zero divisors. (ii) For any noninvertible r ∈ R there exist irreducible elements r1 , . . , rk ∈ R such that r = r1 · · · rk . (iii) For any other factorization r = q1 · · · ql , where q1 , . . , ql are irreducible in R, it holds that l = k, and for any i, 1 ≤ i ≤ k there exist j, 1 ≤ j ≤ k such that qi = ui rj for a certain invertible element ui ∈ R.

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