Heredity (mathematics)

In mathematics, the length of a projective resolution of a module over a ring provides, in a sense, a measure of how “complicated” the module is. ${\displaystyle R}$

A ring is called hereditary if every sub-module of a projective module is projective. This means that every minimum projective resolution of a module stops after two steps. ${\displaystyle R}$ ${\displaystyle R}$

In the case of non- commutative rings, a distinction is made between left and right inheritance: A ring is called left-hereditary if every sub-module of a projective left-hand module is projective. Correspondingly, a ring is called right-hereditary if every sub-module of a projective right-hand module is projective. There are rings that are left-handed but not right-hereditary, and vice versa (see below). ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$

Examples

• Every body is hereditary because all modules (= vector spaces) are free and thus projective. ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle K}$
• Every semi-simple ring is hereditary as every module above the ring is projective.
• Each main ideal ring is hereditary, since here projective modules are free and sub-modules of free modules are also free.
• ${\displaystyle {\begin{pmatrix}\mathbb {Q} &\mathbb {Q} \\0&\mathbb {Z} \end{pmatrix}}:={\Bigl \{}{\begin{pmatrix}a_{1,1}&a_{1,2}\\0&a_{2,2}\end{pmatrix}}\in \mathrm {Mat} (2,\mathbb {Q} )\mid a_{2,2}\in \mathbb {Z} {\Bigr \}}}$ is left-hereditary but not right-hereditary.
• Every path algebra of a quiver is hereditary.

Individual evidence

1. ^ Louis D. Tarmin: Lineare Algebra Modules 2 , Tschampel BuchMat 4.B (2008), ISBN 3-934-67151-9 , definition 1.134.1
2. ↑ Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-125-99841-4 ( Pure and Applied Mathematics 127), definition 2.8.11
3. ↑ Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-125-99841-4 ( Pure and Applied Mathematics 127), exercise 2.8.5