Ideal

A special type of subobject of an algebraic structure. The concept of an ideal first arose in the theory of rings. The name ideal derives from the concept of an ideal number.

For an algebra, a ring or a semi-group $ A $, an ideal $ I $ is a subalgebra, subring or sub-semi-group closed under multiplication by elements of $ A $. Here an ideal is said to be a left (or right) ideal if it is closed under multiplication on the left (or right) by elements of $ A $, that is, if

$$ AI = I \ ( \textrm{ or } IA = A), $$

where

$$ AI = \{ {ab } : {a \in A, b \in I } \} ,\ \ IA = \{ {ba } : {a \in A, b \in I } \} . $$

An ideal that is simultaneously a left ideal and a right ideal (that is, one that is preserved under multiplication by elements of $ A $) is said to be two-sided. These three concepts coincide in the commutative case. Every assertion about left ideals has a corresponding dual assertion about right ideals (subsequent statements will refer only to the "left case" ).

Two-sided ideals in rings and algebras play exactly the same role as do normal subgroups (cf. Normal subgroup) in groups. For every homomorphism $ f: A \rightarrow B $, the kernel $ \mathop{\rm Ker} f $( that is, the set of elements mapped to 0 by $ f $) is an ideal, and conversely every ideal $ I $ is the kernel of some homomorphism. Moreover, an ideal $ I $ determines a unique congruence (in algebra) $ \kappa $ on $ A $ of which it is the zero class, and thus determines the image $ Af $ of the homomorphism $ f $ of which it is the kernel uniquely (up to an isomorphism): $ Af $ is isomorphic to the quotient ring or quotient algebra $ A/ \kappa $, denoted also by $ A/I $. Ideals of multi-operator groups have similar properties in relation to homomorphisms. In a multi-operator $ \Omega $- group $ A $ an ideal is defined to be a normal subgroup of its additive group satisfying the following property: For every $ n $- ary operator $ \omega $, arbitrary elements $ b \in I $ and $ a _ {1} \dots a _ {n} \in A $, the relation

$$ ( a _ {1} {} \dots a _ {n} \omega ) + ( a _ {1} \dots a _ {i - 1 } ( b + a _ {i} ) a _ {i + 1 } \dots a _ {n} \omega ) \in I $$

holds for $ i = 1 \dots n $. (This concept reduces to that of a two-sided ideal for rings and algebras.)

On the other hand, the two-sided ideals of a semi-group do not give a description of all homomorphic images of the semi-group. If a homomorphism $ f $ of a semi-group $ A $ onto a semi-group $ B $ is given, then only in the case where $ B $ is a semi-group with zero it is possible to associate with $ f $ a two-sided ideal in a natural way, namely $ f ^ { - 1 } ( 0) $; however, this association need not determine $ f $ uniquely. Nevertheless, if $ I $ is an ideal of $ A $, then among the quotient semi-groups of $ A $ having the class of $ I $ as an element there exists a maximal one, written $ A/I $( and called the ideal quotient). The elements of this semi-group are the elements of the set $ A \setminus I $ and the ideal $ I $ itself, which is the zero in $ A/I $.

For an arbitrary subset $ X \subset A $ one can define the ideal $ I _ {X} $ generated by $ X $ as the intersection of all ideals that contain $ X $. The set $ X $ is said to be a basis of the ideal $ I _ {X} $. Different bases can generate one and the same ideal. An ideal generated by a single element is said to be a principal ideal.

The intersection, and for semi-groups also the union, of left (two-sided) ideals is again a left (two-sided) ideal. For rings and algebras, the set-theoretical union of ideals need not be an ideal. Let $ I _ {1} $ and $ I _ {2} $ be left or two-sided ideals in a ring (or algebra) $ A $. The sum of the ideals $ I _ {1} $ and $ I _ {2} $ is the ideal $ I _ {1} + I _ {2} = \{ {a + b } : {a \in I _ {1} , b \in I _ {2} } \} $; it is the smallest ideal of $ A $ containing $ I _ {1} $ and $ I _ {2} $. The set of all (left or two-sided) ideals of a ring (or algebra) forms a lattice under the operations of intersection and taking sums. Many classes of rings and algebras are defined by conditions on their ideals or on the lattice of ideals (see Principal ideal ring; Artinian ring; Noetherian ring).

An ideal of the multiplicative semi-group of a ring may or may not be an ideal of the ring. A semi-group $ A $ is a group if and only if $ A $ has no (left or two-sided) ideal other than $ A $. Thus, the abundance of ideals in a semi-group characterizes the degree to which the semi-group differs from a group.

For a $ k $- algebra $ A $( an algebra over a field $ k $), an ideal of the ring $ A $ need not be an ideal of the algebra $ A $. For example, if $ A $ is a $ k $- algebra with zero multiplication, the set of ideals of the ring $ A $ is the set of subgroups of the additive group of $ A $, while the set of ideals of the algebra $ A $ is the set of all subspaces of the vector $ k $- space $ A $. However, when $ A $ is an algebra with identity, these concepts of an ideal coincide. Therefore many results have identical statements for rings and algebras.

A ring not having any two-sided ideal is said to be a simple ring. A ring without proper one-sided ideals is a skew-field. Left ideals of a ring $ A $ may also be defined as submodules of the left $ A $- module $ A $. Some properties of rings remain unchanged when right ideals are substituted for left ideals. For example, the Jacobson radical defined in terms of left ideals is the same as the Jacobson radical defined in terms of right ideals. On the other hand, a left Noetherian ring can fail to be right Noetherian.

The study of ideals in commutative rings is an important part of commutative algebra. With every commutative ring with identity one can associate the topological space $ \mathop{\rm Spec} A $ whose elements are the proper prime ideals of $ A $. There is a one-to-one correspondence between the set of all radicals of ideals of $ A $ and the set of closed subspaces of $ \mathop{\rm Spec} A $.

The concept of an ideal of a field occurs in commutative algebra, more precisely, that of an ideal of a field relative to a ring. Here $ A $ is a commutative ring with identity and without zero divisors, and $ Q $ is the field of fractions of $ A $. An ideal of the field $ Q $ is a non-zero subset $ I \subset Q $ that is a subgroup of the additive group of $ Q $ closed under multiplication by elements of $ A $( that is, $ ab \in I $ whenever $ a \in A $ and $ b \in I $) and such that there exists an element $ q \in Q $ such that $ qI \subset A $. An ideal is said to be an integral ideal if it is contained in $ A $( and then it is an ordinary ideal of $ A $); otherwise it is a fractional ideal.

An ideal of a lattice is a non-empty subset $ I $ of a lattice such that: 1) if $ a, b \in I $, then $ a + b \in I $; and 2) if $ c \leq a \in I $, then $ c \in I $. A dual ideal (or a filter) of a lattice is defined in the dual manner ( $ a, b \in J $ implies $ ab \in J $; $ c \geq a \in J $ implies $ c \in J $). The ideals of a lattice also form a lattice under inclusion. A maximal element of the set of all proper ideals of a lattice is called a maximal ideal. If $ f $ is a homomorphism of a lattice onto a partially ordered set with a zero, then the complete inverse image of the zero is an ideal. It is called the kernel ideal of $ f $. An ideal $ S $ of a lattice $ L $ is said to be a standard ideal if for arbitrary $ a, b \in L $, $ s \in S $, the inequality $ a < b + s $ implies that $ a = x + t $, where $ x \leq b $ and $ t \in S $. Every standard ideal is a kernel ideal. A kernel ideal of a relatively complemented lattice (see Lattice with complements) is standard. An ideal $ I $ is called a prime ideal if $ a \in I $ or $ b \in I $ whenever $ ab \in I $. Each of the following conditions is equivalent to primality for an ideal $ I $ of a lattice $ L $: a) the complement $ A \setminus I $ is a filter; or b) $ I $ is the complete inverse image of zero under some homomorphism of $ L $ onto a two-element lattice. Every maximal ideal of a distributive lattice is prime.

The concept of an ideal in a partially ordered set is not in full agreement with the preceding definition. In fact, instead of 1), a stronger condition is required to hold: For every subset of the ideal, the supremum (join) of the set (if it exists) is also in $ I $.

An ideal object $ A $ of a category with null morphisms is a subobject $ ( U, \mu ) $ of $ A $ such that $ \mu = \mathop{\rm ker} \alpha $ for some morphism $ \alpha : A \rightarrow B $. This ideal can be identified with the set of all monomorphisms that are kernels of some morphism (see also Normal monomorphism). The concept of a co-ideal object of a category is defined in the dual way. The concept of an ideal for $ \Omega $- groups is a special case of that of an ideal object in a category.

A left ideal of a category $ \mathfrak K $ is a class of morphisms containing, with every morphism $ \phi $ of it, all products $ \alpha \phi $ with $ \alpha \in \mathfrak K $, if these are defined in $ \mathfrak K $. Right ideals of a category are defined in the dual way. A two-sided ideal is a class of morphisms that is both a left ideal and a right ideal.

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There is some disagreement about the correct definition of an ideal $ I $ in a partially ordered set $ A $. Instead of the definition given above, some authors would allow $ I $ to be an arbitrary lower set (if $ a \leq b \in I $, then $ a \in I $); others require additionally that $ I $ be directed (if $ a \in I $ and $ b \in I $, then there exists a $ c \in I $ with $ a \leq c $ and $ b \leq c $). The latter definition has the advantage of agreeing with the usual one in the case when $ A $ is a lattice (or a join semi-lattice).

For a Boolean algebra $ A $, a subset $ I $ of $ A $ is an ideal in the lattice-theoretic sense if and only if it is an ideal of the Boolean ring $ A $. It was this equivalence which led M.H. Stone [a1] to extend the use of the term "ideal" from rings to lattices. Since then, the study of ideals has played an important role in lattice theory, and particularly in the theory of distributive lattices.

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