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- W1990402521 abstract "Let R be a ring with identity and let E ij ∈ M n ( R ) be the usual n X n matrix units, where n ≥ 2 and 1≤ i , j ≤ N . Let E n (R) be the subgroup of GL n (R) generated by all T ij ( q where r ∈ R and i ≄ j . For each (two-sided) R -ideal q let E n ( R, q ) be the normal subgroup of E n ( R ) generated by T ij ( q ), where q ∈ q . The subgroup E n ( R, q ) plays an important role in the theory of GL n ( R ). For example, Vaserˇstein has proved that, for a larger class of rings (which includes all commutative rings), every subgroup S of GL n ( R ), when R ∈ and n ≥3, contains the subgroup E n ( R, q 0 ), where q 0 is the R -ideal generated by α ij , r α ij -α jj r ( i ≄ j , r ∈ R ), for all (α ij ) ∈ S . (See [13, Theorem 1].) In addition Vaseršstein has shown that, for the same class of rings, E n ( R, q ) has a simple set of generators when n ≥ 3. Let Ê n ( R, q ) be the subgroup of E n ( R, q ) generated by T ij ( r ) T ij ( q ) T ij (− r ), where r ∈ R , q ∈ q . Then Ê n ( R, q ) = E n ( R, q ), for all q , when R ∈ and n ≥ 3.(See [13, Lemma 8].)" @default.
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- W1990402521 date "1996-01-01" @default.
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- W1990402521 title "On the generators of elementary subgroups of general linear groups" @default.
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- W1990402521 doi "https://doi.org/10.1017/s0017089500031189" @default.
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