Question

Show that no proper left (right) ideal of an algebra with identity can contain an element that has a left (right) inverse.

Short answer

Given that there is an element in the left ideal of the algebra that has a left inverse, we consequently showed that the left ideal contains the complete set of the algebra, contradicting it being a proper subset. Therefore, no proper left ideal can contain an element with a left inverse.

Step-by-step solution

Denote elements in the algebra

Let \(A\) be an algebra with identity \(1\) and \(I\) a left ideal of \(A\). Suppose there exists an element \(a\) in \(I\) such that for some element \(b\) in \(A\), \(ba = 1\).

Apply left multiplicative absorption

For any element \(x\) in \(A\), consider the product \(ax\). Since \(I\) is a left ideal, this means that \(ax\) is also an element of \(I\).

Manipulate the product to reveal identity

However, by the associative property of multiplication in an algebra, we have that \(ax = (ba)x = b(ax) = b1 = b\), where \(b1\) is simply \(b\) as multiplication by identity leaves the element unchanged.

Ideal is the whole algebra

This implies that for every element \(x\) in \(A\), there exists an element \(b\) in \(I\), meaning \(I\) contains every element in \(A\), thus \(I\) must be the entire algebra \(A\). This contradicts the assumption that \(I\) is a proper left ideal, i.e., a proper subset of \(A\), and hence completes the proof.

Key concepts

Left Ideal

In an algebra with identity, a left ideal is a special subset that behaves well with the operations of the algebra. It allows us to consistently multiply elements from outside the subset on the left without leaving the subset.

• A left ideal in an algebra means if you take any element from the ideal and multiply it on the left by an element from the whole algebra, the result still remains inside the ideal.
• This property is fundamental because it allows for the structure to be preserved under multiplication.

For the task to show no proper left ideal contains an element with a left inverse, assume there is such an element with a left inverse. This would mean the entire algebra can be expressed through the left ideal, which contradicts the proper subset definition.

Right Ideal

Right ideals, similar to left ideals, are subsets within an algebra that also respect multiplication, but on the right-hand side. This means if you multiply a member of the right ideal on the right by any element in the algebra, the result stays within the ideal.

• For example, in a right ideal, if you have element \( a \) from the ideal, and \( x \) from the algebra, \( ax \) remains in the ideal.
• This consistency is critical for maintaining the order and structure within algebraic operations.

Demonstrating that a proper right ideal cannot contain an element with a right inverse follows similarly as with left ideals. If such an element existed, the ideal would cover the entire algebra, contradicting its status as a proper subset.

Associative Property

The associative property is a cornerstone in the structure of many mathematical operations, including algebras. It ensures that the order in which operations are performed doesn’t change the result.

• This means for any elements \( a, b, \) and \( c \) in an algebra, the relationship \( (ab)c = a(bc) \) always holds.
• Such a property guarantees that when grouping operations, computations remain predictable and valid.

In the problem demonstrated in the solution, the associative property provides the crucial step to manipulate expressions. It showed how multiplying \( (ba)x \) as \( b(ax) \) simplified to \( b \), reinforcing the identity property and thereby implying the entire algebra would be covered if a left or right inverse existed within a proper ideal.

Inverse Elements

Inverse elements play an important role in ensuring that elements of an algebra can "cancel out" each other, returning to the identity element. A left inverse refers to an element \( b \) such that when it multiplies another element \( a \) from the left, it returns the identity element of the algebra.

• An element \( a \) with left inverse \( b \) satisfies \( ba = 1 \), where \( 1 \) is the identity.
• Similarly, a right inverse satisfies \( ab = 1 \).

If an element within a proper ideal has a left or right inverse, it suggests this element leads to every other element, making the ideal equivalent to the whole algebra. This fundamentally contradicts the nature of a proper subset.