Idempotents from algebraic elements

Let x be an element of an algebra over a field K, algebraic over K and not nilpotent. Write a nonzero polynomial annihilating x and divisible by X as X ^ k * g with k ≥ 1 and g (0) ≠ 0. As X ^ k and g are coprime, a * X ^ k + b * g = 1 for some polynomials a and b, and e = a (x) * x ^ k is an idempotent of the form p (x) * x. It is nonzero, for otherwise b (x) * g (x) = 1 would force x ^ k = 0.

A left ideal consisting of nilpotent elements lies in the Jacobson radical, so a left ideal of algebraic elements lies in the radical exactly when its only idempotent is 0. Every element of a finite-dimensional left ideal is algebraic, so this applies to finite-dimensional left ideals. In a semisimple algebra, meaning one with zero Jacobson radical, every nonzero finite-dimensional left ideal contains a nonzero idempotent. This is the source of the idempotents in the structure theory of separating ideals of Banach algebras.

Main results

• IsAlgebraic.exists_isIdempotentElem_aeval_mul_ne_zero: an algebraic element x that is not nilpotent has a multiple p (x) * x that is a nonzero idempotent.
• Submodule.exists_isIdempotentElem_ne_zero_mem_of_isAlgebraic: a left ideal of algebraic elements not contained in the Jacobson radical contains a nonzero idempotent.
• Submodule.exists_isIdempotentElem_ne_zero_mem: a finite-dimensional left ideal not contained in the Jacobson radical contains a nonzero idempotent.
• Submodule.le_jacobson_iff_forall_isIdempotentElem_eq_zero_of_isAlgebraic: a left ideal of algebraic elements lies in the Jacobson radical exactly when its only idempotent is 0.
• Submodule.le_jacobson_iff_forall_isIdempotentElem_eq_zero: a finite-dimensional left ideal lies in the Jacobson radical exactly when its only idempotent is 0.
• Submodule.exists_isIdempotentElem_ne_zero_mem_of_jacobson_eq_bot: in an algebra with zero Jacobson radical, every nonzero finite-dimensional left ideal contains a nonzero idempotent.

References

• H. G. Dales, Banach Algebras and Automatic Continuity, §1.5, for the idempotents of finite-dimensional ideals, and §5.2, for their use in the structure of separating ideals.
• T. Y. Lam, A First Course in Noncommutative Rings, §4: nil one-sided ideals lie in the Jacobson radical.

theorem IsAlgebraic.exists_isIdempotentElem_aeval_mul_ne_zero {K : Type u_1} {B : Type u_2} [Field K] [Ring B] [Algebra K B] {x : B} (hx : IsAlgebraic K x) (hnil : ¬IsNilpotent x) : ∃ (p : Polynomial K), IsIdempotentElem ((Polynomial.aeval x) p * x) ∧ (Polynomial.aeval x) p * x ≠ 0

Idempotents from algebraic elements. If x is algebraic over a field K and not nilpotent, then for some polynomial p over K the element p (x) * x is a nonzero idempotent.

theorem Submodule.exists_isIdempotentElem_ne_zero_mem_of_isAlgebraic {K : Type u_1} {B : Type u_2} [Field K] [Ring B] [Algebra K B] (J : Submodule K B) (hJ : ∀ (b : B) {y : B}, y ∈ J → b * y ∈ J) (halg : ∀ y ∈ J, IsAlgebraic K y) (hrad : ¬J ≤ restrictScalars K (Ring.jacobson B)) : ∃ e ∈ J, IsIdempotentElem e ∧ e ≠ 0

Idempotents in left ideals outside the radical. A left ideal J of an algebra B over a field, given as a scalar submodule closed under left multiplication, whose elements are algebraic and which is not contained in the Jacobson radical of B contains a nonzero idempotent.

theorem Submodule.exists_isIdempotentElem_ne_zero_mem {K : Type u_1} {B : Type u_2} [Field K] [Ring B] [Algebra K B] (J : Submodule K B) [FiniteDimensional K ↥J] (hJ : ∀ (b : B) {y : B}, y ∈ J → b * y ∈ J) (hrad : ¬J ≤ restrictScalars K (Ring.jacobson B)) : ∃ e ∈ J, IsIdempotentElem e ∧ e ≠ 0

Idempotents in finite-dimensional left ideals. A finite-dimensional left ideal J of an algebra B over a field, given as a scalar submodule closed under left multiplication, that is not contained in the Jacobson radical of B contains a nonzero idempotent.

theorem Submodule.le_jacobson_iff_forall_isIdempotentElem_eq_zero_of_isAlgebraic {K : Type u_1} {B : Type u_2} [Field K] [Ring B] [Algebra K B] (J : Submodule K B) (hJ : ∀ (b : B) {y : B}, y ∈ J → b * y ∈ J) (halg : ∀ y ∈ J, IsAlgebraic K y) : J ≤ restrictScalars K (Ring.jacobson B) ↔ ∀ e ∈ J, IsIdempotentElem e → e = 0

A left ideal of algebraic elements of an algebra over a field, given as a scalar submodule closed under left multiplication, lies in the Jacobson radical exactly when its only idempotent is 0.

theorem Submodule.le_jacobson_iff_forall_isIdempotentElem_eq_zero {K : Type u_1} {B : Type u_2} [Field K] [Ring B] [Algebra K B] (J : Submodule K B) [FiniteDimensional K ↥J] (hJ : ∀ (b : B) {y : B}, y ∈ J → b * y ∈ J) : J ≤ restrictScalars K (Ring.jacobson B) ↔ ∀ e ∈ J, IsIdempotentElem e → e = 0

A finite-dimensional left ideal of an algebra over a field, given as a scalar submodule closed under left multiplication, lies in the Jacobson radical exactly when its only idempotent is 0.

theorem Submodule.exists_isIdempotentElem_ne_zero_mem_of_jacobson_eq_bot {K : Type u_1} {B : Type u_2} [Field K] [Ring B] [Algebra K B] (J : Submodule K B) [FiniteDimensional K ↥J] (hJ : ∀ (b : B) {y : B}, y ∈ J → b * y ∈ J) (hB : Ring.jacobson B = ⊥) (hJ0 : J ≠ ⊥) : ∃ e ∈ J, IsIdempotentElem e ∧ e ≠ 0

In an algebra over a field with zero Jacobson radical, every nonzero finite-dimensional left ideal, given as a scalar submodule closed under left multiplication, contains a nonzero idempotent.