By N. A. Sanin
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Algebra, 13:53–70, 2004. In Russian.  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  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.  L. B. Beasley, A. E. -G. -Z. Song. Linear transformations preserving the Grassmannian over Mn (Z+ ). , 393:39–46, 2004.  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.