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| Let $R$ be a ring. For fixed positive integer $n$, $R$ is said to be left $n$-semihereditary in case every $n$-generated left ideal is projective. $R$ is said to be weakly $n$-semihereditary if each $n$-generated left (and/or right) ideal is flat. Some properties of $n$-semihereditary rings, respectively, weakly $n$-semihereditary rings and $n$-coherent rings are investigated. It is also proved that $R$ is left $n$-semihereditary if and only if it is left $n$-coherent and weakly $n$-semihereditary, if and only if the ring of $n\times n$ matrices over $R$ is left 1-semihereditary if and only if the class of all $n$-flat right $R$-modules form the torsion-free class of a torsion theory. $R$ is left semihereditary if and only if it is left $n$-semihereditary for all positive integers $n$. |
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| Keywords:$n$-semihereditary ring, weakly $n$-semihereditary ring, $n$-coherent ring |
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