# Why is an associative semisimple algebra the direct sum of minimal left ideals?

I'm reading Clausen, Fast Fourier Transforms, and he states (p. 34):

Let A be an associative algebra.
A is semisimple $\Leftrightarrow$ A is a direct sum of minimal left ideals.

Can anyone explain why?

Followup question: since A is semisimple, its left A-modules are completely reducible, i.e. every left A-module is the direct sum of simple left A-modules. Is there an easy correspondence between the simple A-modules and the left ideals of A?

-

– YACP Nov 23 '12 at 23:46

## 1 Answer

The equivalence between these conditions in an associative ring $R$ are covered in almost every book with noncommutative rings:

1. Every right ideal of $R$ is a direct summand. (Same for left ideals)

2. $R$ is a direct sum of simple right ideals. (Same for left ideals)

3. Every $R$ module is a direct sum of simple $R$ modules.

As for the followup question (usually it's better to post these separately, but this one is easy enough to cover here) the simple right $R$ modules will all be represented by the simple right ideals. There may be duplicate copies though!

Here is how to see it. If $M$ is a simple right $R$ module, then there is a homomorphism $R\rightarrow M$ that is onto. Thus $M\cong R/T$ where $T$ is a maximal right ideal. But since $R$ is semisimple $R\cong T\oplus N$, and you can verify that $N$ is a minimal right ideal. Finally, $M\cong N$.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 16, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8927532434463501, "perplexity_flag": "head"}