The optimal version of Hua's fundamental theorem of geometry by Peter Semrl

By Peter Semrl

Hua's basic theorem of geometry of matrices describes the overall kind of bijective maps at the area of all m�n matrices over a department ring D which shield adjacency in either instructions. influenced through numerous purposes the writer experiences a protracted status open challenge of attainable advancements. There are 3 ordinary questions. will we change the belief of protecting adjacency in either instructions through the weaker assumption of maintaining adjacency in a single path simply and nonetheless get a similar end? do we sit back the bijectivity assumption? will we receive the same end result for maps performing among the areas of oblong matrices of alternative sizes? A department ring is expounded to be EAS whether it is no longer isomorphic to any right subring. For matrices over EAS department earrings the writer solves all 3 difficulties concurrently, hence acquiring the optimum model of Hua's theorem. when it comes to normal department jewelry he will get such an optimum consequence just for sq. matrices and offers examples exhibiting that it can't be prolonged to the non-square case

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0 0 ... 0 0 ... 0 0 ... .. . . 0 0 ... 0 0 ... .. . . k = 1, . . , n. Here, Sk has exactly k nonzero entries in the second column and exactly the first k + 1 entries of the first column of Pk are equal to 1. From Sk ≤ E11 + . . + Ek+1,k+1 , (30), and the induction hypothesis we get that ⎡ ⎤ 1 0 0 ... 0 0 ... 0 ⎢0 1 ∗ . . ak 0 . . 0⎥ ⎢ ⎥ ⎢ ⎥ ϕ(Sk ) = ⎢0 0 0 . . 0 0 . . 0⎥ , ⎢ .. .. . .. . ⎥ ⎣. . ⎦ . . 0 0 0 ... 0 0 ... 0 where the entry ak in the (2, k + 1)-position is P L( t e1 ) and Pk ≤ E11 + .

0 ∗ 0 ... 0 0 0 0 ... 0 and consequently, ⎛⎡ ⎤ ⎤⎞ ⎡ 1 0 ... 0 1 ∗ ... ∗ ⎜⎢0 0 . . 0⎥⎟ ⎢∗ 0 . . 0⎥ ⎜⎢ ⎥ ⎥⎟ ⎢ ξ ⎜⎢ . . ⎟ = ⎢. ⎥ . . ⎥ ⎝⎣ .. ⎦⎠ ⎣ .. . . ⎦ ∗ 0 ... 0 0 0 ... 0 Similarly, ⎛⎡ (30) ⎤⎞ ⎡ ⎤ 1 0 ... 0 1 ∗ ... ∗ ⎜⎢∗ 0 . . 0⎥⎟ ⎢0 0 . . 0⎥ ⎜⎢ ⎥⎟ ⎢ ⎥ ξ ⎜⎢ . . ⎟ = ⎢. ⎥ . . ⎥ ⎝⎣ .. ⎦⎠ ⎣ .. . . ⎦ ∗ 0 ... 0 0 0 ... 0 Because ξ preserves rank one idempotents and adjacency, each subset P L( t y) ⊂ Pn (D) is mapped either into some P L( t w) or some P R(x), and the same is true for each subset P R(z) ⊂ Pn (D).

0 0 ... .. . . 0 0 ... 0 0 ... .. . . k = 1, . . , n. Here, Sk has exactly k nonzero entries in the second column and exactly the first k + 1 entries of the first column of Pk are equal to 1. From Sk ≤ E11 + . . + Ek+1,k+1 , (30), and the induction hypothesis we get that ⎡ ⎤ 1 0 0 ... 0 0 ... 0 ⎢0 1 ∗ . . ak 0 . . 0⎥ ⎢ ⎥ ⎢ ⎥ ϕ(Sk ) = ⎢0 0 0 . . 0 0 . . 0⎥ , ⎢ .. .. . .. . ⎥ ⎣. . ⎦ . . 0 0 0 ... 0 0 ... 0 where the entry ak in the (2, k + 1)-position is P L( t e1 ) and Pk ≤ E11 + .

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