Difference between revisions of "Uniserial ring"
From Encyclopedia of Mathematics
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A ring all indecomposable one-sided ideals of which have a unique composition series, and which splits into the direct sum of primary rings. Omission of the latter requirement leads to the definition of a generalized uniserial ring, also called a serial ring. Every generalized uniserial ring is a semi-chain ring both on the left and the right (see [[Semi-chain module|Semi-chain module]]; [[Semi-chain ring|Semi-chain ring]]). Every module over a generalized uniserial ring splits into the direct sum of cyclic submodules. A ring is a generalized uniserial ring if and only if all left modules over it are semi-chain modules. An example of a uniserial ring is the ring of upper-triangular matrices over a skew-field. | A ring all indecomposable one-sided ideals of which have a unique composition series, and which splits into the direct sum of primary rings. Omission of the latter requirement leads to the definition of a generalized uniserial ring, also called a serial ring. Every generalized uniserial ring is a semi-chain ring both on the left and the right (see [[Semi-chain module|Semi-chain module]]; [[Semi-chain ring|Semi-chain ring]]). Every module over a generalized uniserial ring splits into the direct sum of cyclic submodules. A ring is a generalized uniserial ring if and only if all left modules over it are semi-chain modules. An example of a uniserial ring is the ring of upper-triangular matrices over a skew-field. | ||
Latest revision as of 16:40, 1 May 2014
A ring all indecomposable one-sided ideals of which have a unique composition series, and which splits into the direct sum of primary rings. Omission of the latter requirement leads to the definition of a generalized uniserial ring, also called a serial ring. Every generalized uniserial ring is a semi-chain ring both on the left and the right (see Semi-chain module; Semi-chain ring). Every module over a generalized uniserial ring splits into the direct sum of cyclic submodules. A ring is a generalized uniserial ring if and only if all left modules over it are semi-chain modules. An example of a uniserial ring is the ring of upper-triangular matrices over a skew-field.
How to Cite This Entry:
Uniserial ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniserial_ring&oldid=32110
Uniserial ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniserial_ring&oldid=32110
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article