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Zero sets—consequences for primitive near-rings

1982
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Proceedings of the Edinburgh Mathematical Society
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Proof. By Corollary 2 of 1.1, Z[(0: Z,)]2Z[(0: S)], and since S-(0:S) = {0}, S c 2 [ ( 0 : S)]. By 1.2, Z[(0: Z J^Z j and the result follows. Corollary 2. // Z t and Z 2 are bofh Z-closed subsets of V, then Z x = Z 2 i/, and onfy if, (0:Z 1 ) = (0:Z 2 ). This follows since, if (0: Z t ) = (0: Z 2 ), then by 1.2, Z x = Z[(0: Z x )] = Z[(0: Z 2 )] = Z 2 . Corollary 2 puts zero sets and their annihilators in one-one correspondence. The next proposition shows this correspondence is lattice

doi:10.1017/s0013091500004132
fatcat:ek4z37nenraqteltrk5zqsw4uq