Minimal ideal
A minimal element of the
partially ordered set
of ideals of a given type of some algebraic system.
Since the ordering on the set of ideals is defined
by the inclusion relation, a minimal ideal is an ideal not
containing ideals of the same type different
from itself. For multi-operator groups (in particular,
rings)
and for
lattices,
in contrast to
semi-groups,
it is always assumed that the partially ordered set of ideals
does not contain the zero ideal. If no class of ideals
is specifically mentioned, a minimal ideal is taken to be
minimal in the set of all (non-zero) two-sided ideals.
A minimal two-sided ideal, if one exists, in a
semi-group
is unique and is the smallest two-sided ideal: it is called the
kernel of the semi-group
 .
Not every semi-group has a kernel (for example, an infinite
monogenic semi-group)
but, for example, the kernel exists in any finite
semi-group. The kernel is an ideally-simple semi-group (see
Simple semi-group).
If the kernel of a semi-group
is a
group,
then
is called a
homogroup.
A semi-group
is a homogroup if and only if there is an element
in
which is divisible on the left and right by any element of
(that is,
for any
 );
in this case the kernel consists of all such elements.
For example, every finite commutative semi-group is a homogroup.
If a semi-group
has a minimal left ideal
 ,
then for any
the product
is also a minimal left ideal, moreover, every minimal
left ideal can be obtained in this way. Every minimal left
ideal is a left simple semi-group. In a semi-group with minimal left
ideals every left ideal contains a minimal left ideal, and the union of
all minimal left ideals (which are pairwise disjoint) is
the kernel of the semi-group. If a semi-group
has a minimal left ideal
and a minimal right ideal
 ,
then
is a subgroup in
and
 ,
 ,
where
is the identity of this subgroup; the product
coincides with the kernel of
and is, in this case, a
completely-simple semi-group.
For semi-groups with a zero, the interest is in the consideration
of non-zero ideals, and a minimal element in the
corresponding partially ordered set of ideals is called a
 -minimal
(left,
right,
two-sided)
ideal.
The properties of
 -minimal
ideals are in many ways similar to the properties of
minimal ideals, with some natural restrictions. For example, a
 -minimal
two-sided ideal is not necessarily unique and need not be a
 -simple
semi-group; it may be a semi-group with zero multiplication (see
Nilpotent semi-group).
The union of all
 -minimal
left ideals (respectively,
 -minimal
right ideals) is a semi-group with zero, called its
left
(respectively,
right)
socle
(by definition, the
socle
is equal to zero if there are no corresponding
 -minimal
ideals in the semi-group). A semi-group coincides with its
left and right socles if and only if it is an
 -direct
union of completely-simple semi-groups and semi-groups with zero multiplication.
The consideration of minimal ideals and
 -minimal
ideals plays an essential role in the structure theory of
a number of important classes of semi-groups (see, for example,
Completely-simple semi-group;
Regular semi-group,
and also
,
§§ 2.5, 2.7, Chapt. 6, §§ 7.7, 8.2, 8.3;
,
Chapt. V).
L.N. Shevrin
Rings (like semi-groups) need not have minimal ideals (the simplest example
is the ring of integers) and a minimal ideal in a ring, if
it exists, need not be unique. The sum of all (left, right,
two-sided) minimal ideals in a ring is called the
(left,
right,
two-sided)
socle of the ring.
An
Artinian ring,
obviously, has a non-zero socle. The presence of minimal ideals in a
primitive ring
makes it close to a matrix ring in the following
sense: A primitive ring with a non-zero socle is isomorphic to
a dense subring of the ring of all linear transformations
of some vector space over a skew-field, containing all transformations of finite rank
.
V.E. Govorov
References for the above sections are given below.
References| [1] |
A.H. Clifford,
G.B. Preston,
"Algebraic theory of semi-groups"
, 1–2
, Amer. Math. Soc.
(1961–1967) | | [2] |
E.S. Lyapin,
"Semigroups"
, Amer. Math. Soc.
(1974)
(Translated from Russian) | | [3] |
N. Jacobson,
"Structure of rings"
, Amer. Math. Soc.
(1956) |
CommentsIn commutative ring theory, and also in the theory
of distributive lattices, the study of minimal prime ideals (i.e.
minimal elements in the ordered set of prime ideals) plays
an important part. To a large extent this is simply
the order-theoretic dual of the study of maximal ideals of these structures (see
Maximal ideal),
but the parallel is not exact — for instance, the space
of minimal prime ideals of a distributive lattice is always
Hausdorff but need not be compact, whereas the space of
maximal ideals is always compact but need not be Hausdorff.
References| [a1] |
M. Henriksen,
M. Jerison,
"The space of minimal prime ideals of a commutative ring"
Trans. Amer. Math. Soc.
, 115
(1965)
pp. 110–130 | | [a2] |
H. Simmons,
"Reticulated rings"
J. Algebra
, 66
(1980)
pp. 169–192 | | [a3] |
S.-H. Sun,
"A localic approach to minimal prime spectra"
Math. Proc. Cambridge Philos. Soc.
, 103
(1988)
pp. 47–53 |
This text originally appeared in Encyclopaedia of Mathematics
- ISBN 1402006098
|