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How to work without an identity element when you clearly need it

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Quick description

Several kinds of rings present a structure weaker than having an identity element, but "near to it" in some local, categorical or loose sense:

  • Idempotent rings

  • Firm rings

  • s-unital rings

  • Rings with local units

  • Rings with enough idempotents

There also exist some other techniques:

  • Peirce decomposition

  • Unitizations

Prerequisites

Example 1

General discussion

See also:

Decompose your ring using idempotents

If you don't have an identity element, a set of local units is more than enough

To make a function nicer without changing it much, convolve with an approximate delta function

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