- #1

Geofleur

Science Advisor

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## Main Question or Discussion Point

In the chapter on Algebras in Hassani's mathematical physics text, left ideals are defined as follows:

Let ## \mathcal{A} ## be an algebra. A subspace ## \mathcal{B} ## of ## \mathcal{A} ## is called a left ideal of ## \mathcal{A} ## if it contains ## \mathbf{a}\mathbf{b} ## for all ## \mathbf{a}\in \mathcal{A} ## and ## \mathbf{b} \in \mathcal{B} ##.

He then defines a minimal left ideal:

A left ideal ## \mathcal{M} ## of an algebra ## \mathcal{A} ## is called minimal if every left ideal of ## \mathcal{A} ## contained in ## \mathcal{M} ## coincides with ## \mathcal{M} ##.

Here is where I am confused. The set containing only the zero vector is a subspace of any vector space, because ## \alpha \mathbf{0} + \beta \mathbf{0} = \mathbf{0} ## for any scalars ## \alpha ## and ## \beta ##. Moreover, the set containing the zero vector is a subalgebra of any algebra, because ## \mathbf{0} \mathbf{0} = \mathbf{0} ##. In fact, the "zero set" is a left ideal of any algebra, because ## \mathbf{a} \mathbf{0} = \mathbf{0} ## for any ##\mathbf{a} \in \mathcal{A} ##. But then the only minimal left ideal is just the zero set, because every left ideal has the zero vector as an element. This conclusion would make the whole concept of minimal ideals rather uninteresting. Am I going wrong somewhere here?

Let ## \mathcal{A} ## be an algebra. A subspace ## \mathcal{B} ## of ## \mathcal{A} ## is called a left ideal of ## \mathcal{A} ## if it contains ## \mathbf{a}\mathbf{b} ## for all ## \mathbf{a}\in \mathcal{A} ## and ## \mathbf{b} \in \mathcal{B} ##.

He then defines a minimal left ideal:

A left ideal ## \mathcal{M} ## of an algebra ## \mathcal{A} ## is called minimal if every left ideal of ## \mathcal{A} ## contained in ## \mathcal{M} ## coincides with ## \mathcal{M} ##.

Here is where I am confused. The set containing only the zero vector is a subspace of any vector space, because ## \alpha \mathbf{0} + \beta \mathbf{0} = \mathbf{0} ## for any scalars ## \alpha ## and ## \beta ##. Moreover, the set containing the zero vector is a subalgebra of any algebra, because ## \mathbf{0} \mathbf{0} = \mathbf{0} ##. In fact, the "zero set" is a left ideal of any algebra, because ## \mathbf{a} \mathbf{0} = \mathbf{0} ## for any ##\mathbf{a} \in \mathcal{A} ##. But then the only minimal left ideal is just the zero set, because every left ideal has the zero vector as an element. This conclusion would make the whole concept of minimal ideals rather uninteresting. Am I going wrong somewhere here?