# Idempotence

**Idempotence** is the property of an element \( e \) of a ring (or, more generally, of any structure with an associative multiplication) satisfying \( e^2 = e \): multiplying the element by itself changes nothing. Such an element is called an **idempotent**. The condition is equivalent to \( e(1-e) = 0 \) in any ring with identity, and it makes the multiplication by \( e \) act like a projection: applying it once fixes everything it will ever fix.<sup>[1](https://ocw.mit.edu/courses/res-18-012-algebra-ii-student-notes-spring-2022/mit18_702s22_full_lec.pdf)</sup>

| Key fact | Detail |
|---|---|
| Defining equation | \( e^2 = e \), equivalently \( e(1-e) = 0 \) in a ring with 1<sup>[1](https://ocw.mit.edu/courses/res-18-012-algebra-ii-student-notes-spring-2022/mit18_702s22_full_lec.pdf)</sup> |
| Trivial examples | 0 and 1 are idempotent in every ring; in a field these are the only ones<sup>[1](https://ocw.mit.edu/courses/res-18-012-algebra-ii-student-notes-spring-2022/mit18_702s22_full_lec.pdf)</sup> |
| Complement | If \( e \) is idempotent, so is \( 1-e \), and the two are orthogonal<sup>[1](https://ocw.mit.edu/courses/res-18-012-algebra-ii-student-notes-spring-2022/mit18_702s22_full_lec.pdf)</sup> |
| Ring splitting | \( R \cong R_1 \times \cdots \times R_k \) exactly when there are mutually orthogonal central idempotents \( e_1 + \cdots + e_k = 1 \)<sup>[2](https://www.math.ucla.edu/~sharifi/notes/algebra-ch13.html)</sup> |
| Boolean algebra | In a commutative ring, the idempotents form a Boolean algebra with meet \( ee' \) and join \( e + e' - ee' \)<sup>[3](https://www.math.stonybrook.edu/~jstarr/M535s26/M535s26field.pdf)</sup> |
| Geometry | Clopen subsets of \( \mathrm{Spec}(R) \) correspond one-to-one to idempotents of \( R \)<sup>[4](https://stacks.math.columbia.edu/tag/04PN)</sup> |
| Matrix form | An idempotent matrix \( A \) (\( A^2 = A \)) has only the eigenvalues 0 and 1, is diagonalizable, and satisfies \( \operatorname{tr}(A) = \operatorname{rank}(A) \)<sup>[5](https://pws.npru.ac.th/sartthong/data/files/Matrix%20Algebra%20theory%20computations%20and%20applications%20in%20statistics.pdf)</sup> |

## Basic properties

The equation \( e^2 = e \) rearranges to \( e(1-e) = 0 \), so an idempotent is a zero divisor unless it is 0 or 1. Two immediate consequences follow. First, \( 1 - e \) is again idempotent whenever \( e \) is, because \( (1-e)^2 = 1 - 2e + e^2 = 1 - e \). Second, in a field (or any integral domain) the product \( e(1-e) = 0 \) forces \( e = 0 \) or \( e = 1 \), so fields, integral domains, and local rings have no idempotents beyond the two trivial ones.<sup>[1](https://ocw.mit.edu/courses/res-18-012-algebra-ii-student-notes-spring-2022/mit18_702s22_full_lec.pdf)</sup><sup> • </sup><sup>[3](https://www.math.stonybrook.edu/~jstarr/M535s26/M535s26field.pdf)</sup>

Products of rings supply the standard nontrivial examples. In \( R \times S \), the elements \( e_1 = (1_R, 0) \) and \( e_2 = (0, 1_S) \) are idempotents that are neither 0 nor 1.<sup>[1](https://ocw.mit.edu/courses/res-18-012-algebra-ii-student-notes-spring-2022/mit18_702s22_full_lec.pdf)</sup> Ring homomorphisms preserve idempotence: if \( u : R \to S \) is a morphism of commutative rings and \( e \) is idempotent in \( R \), then \( u(e) \) is idempotent in \( S \).<sup>[3](https://www.math.stonybrook.edu/~jstarr/M535s26/M535s26field.pdf)</sup>

**Orthogonality and primitiveness.** Two idempotents \( e, f \) are *orthogonal* if \( ef = fe = 0 \).<sup>[2](https://www.math.ucla.edu/~sharifi/notes/algebra-ch13.html)</sup> Any finite sum of pairwise orthogonal idempotents is itself idempotent.<sup>[2](https://www.math.ucla.edu/~sharifi/notes/algebra-ch13.html)</sup> A nonzero idempotent is *primitive* if it cannot be written as a sum of two nonzero orthogonal idempotents.<sup>[6](https://www.sciencedirect.com/science/article/pii/S0304020808701981)</sup> A *central* idempotent is one that commutes with every element of the ring; a primitive central idempotent cannot be split as a sum of two nonzero central orthogonal idempotents.<sup>[7](https://www.math.unl.edu/~jkettinger2/Finite-Dimensional%20Algebras.pdf)</sup>

## Idempotents and decompositions of rings

The structural importance of idempotents comes from the correspondence between central idempotents and product decompositions. A ring \( R \) decomposes as a direct product \( R_1 \times R_2 \times \cdots \times R_k \) if and only if there exist mutually orthogonal central idempotents \( e_1, \ldots, e_k \) in the center \( Z(R) \) with \( e_1 + \cdots + e_k = 1 \); the factors may be taken to be \( R_i = Re_i \).<sup>[2](https://www.math.ucla.edu/~sharifi/notes/algebra-ch13.html)</sup> Each \( Re_i \) is a two-sided ideal and a ring in its own right with identity \( e_i \), though for \( k \ge 2 \) it is not a subring of \( R \) in the unital sense, since \( e_i \neq 1 \).<sup>[2](https://www.math.ucla.edu/~sharifi/notes/algebra-ch13.html)</sup> The same construction splits modules: for a left \( R \)-module \( M \), one has \( M = e_1 M \oplus \cdots \oplus e_k M \).<sup>[2](https://www.math.ucla.edu/~sharifi/notes/algebra-ch13.html)</sup>

In a finite-dimensional algebra, writing \( 1 = f_1 + f_2 + \cdots + f_n \) as a sum of pairwise orthogonal central idempotents, with \( n \) as large as possible, determines the \( f_i \) uniquely; these primitive central idempotents decompose the algebra into indecomposable two-sided ideals.<sup>[7](https://www.math.unl.edu/~jkettinger2/Finite-Dimensional%20Algebras.pdf)</sup> This "technique of idempotents" and the associated Peirce decompositions are widely applied in the study of finite-dimensional algebras.<sup>[8](https://encyclopediaofmath.org/wiki/Rings_and_algebras)</sup>

**Boolean algebra structure.** For a commutative ring \( R \), the set \( \mathrm{Idem}(R) \) of idempotents carries meet \( e \wedge e' = ee' \) and join \( e \vee e' = e + e' - ee' \), under which it is a [Boolean algebra](https://www.edgepedia.org/boolean-algebra) with join identity 0 and meet identity 1.<sup>[3](https://www.math.stonybrook.edu/~jstarr/M535s26/M535s26field.pdf)</sup> When \( R \) has a finite primitive decomposition \( \Sigma \), every idempotent is a sum \( \sum_{e \in I} e \) over a unique subset \( I \subseteq \Sigma \), and \( \mathrm{Idem}(R) \) is isomorphic to the power set Boolean algebra \( P(\Sigma) \).<sup>[3](https://www.math.stonybrook.edu/~jstarr/M535s26/M535s26field.pdf)</sup> A unital [Boolean ring](https://www.edgepedia.org/boolean-ring), that is, a ring in which every element is idempotent, is called a Boolean algebra.<sup>[9](https://www.ww2000w.numericana.com/answer/rings.htm)</sup>

## Idempotents in algebraic geometry and lifting

The prime spectrum \( \mathrm{Spec}(R) \) of a commutative ring records idempotents as topology. For an idempotent \( e \), the standard open sets satisfy \( \mathrm{Spec}(R) = D(e) \amalg D(1-e) \), because the image of \( e \) in every residue field is either 0 or 1.<sup>[4](https://stacks.math.columbia.edu/tag/04PN)</sup> Conversely, every subset of \( \mathrm{Spec}(R) \) that is both open and closed equals \( D(e) \) for a unique idempotent \( e \), giving a one-to-one correspondence between clopen subsets and idempotents.<sup>[4](https://stacks.math.columbia.edu/tag/04PN)</sup> Correspondingly, the spectrum of a product \( R_1 \times R_2 \) is the disjoint union of the two spectra.<sup>[10](https://citeseerx.ist.psu.edu/document?doi=fccb92ca6dd3edb2ad17f17ac92ff628fee42ce5&repid=rep1&type=pdf)</sup>

Idempotents also behave well under passage to quotient rings. If \( I \) is a locally nilpotent ideal of \( R \), every idempotent of \( R/I \) lifts to an idempotent of \( R \); when \( R \) is commutative, the map \( R \to R/I \) gives a bijection on idempotents.<sup>[11](https://mathweb.tifr.res.in/~arvind/algebra/Lectures_algebra.pdf)</sup> This lifting lemma underlies the product decomposition of Artinian rings, where \( R/\mathrm{rad}(R) \) is semisimple and its complete set of orthogonal primitive idempotents is lifted back to \( R \).<sup>[11](https://mathweb.tifr.res.in/~arvind/algebra/Lectures_algebra.pdf)</sup>

## Idempotent matrices and projections

A square matrix \( A \) with \( A^2 = A \) is an idempotent matrix. Its eigenvalues satisfy \( \lambda^2 = \lambda \), so they are only 0 and 1; the eigenspaces split the ambient vector space as \( V = V_1 \oplus V_0 \), which is the linear-algebra picture of the same decomposition idea.<sup>[1](https://ocw.mit.edu/courses/res-18-012-algebra-ii-student-notes-spring-2022/mit18_702s22_full_lec.pdf)</sup> The number of eigenvalues equal to 1 is the rank, so \( \operatorname{tr}(A) = \operatorname{rank}(A) \); an idempotent matrix is diagonalizable, and \( \operatorname{rank}(I - A) = n - \operatorname{rank}(A) \) for an \( n \times n \) matrix.<sup>[5](https://pws.npru.ac.th/sartthong/data/files/Matrix%20Algebra%20theory%20computations%20and%20applications%20in%20statistics.pdf)</sup> Every idempotent matrix is singular unless it is the identity matrix.<sup>[5](https://pws.npru.ac.th/sartthong/data/files/Matrix%20Algebra%20theory%20computations%20and%20applications%20in%20statistics.pdf)</sup>

A symmetric idempotent matrix is called a projection matrix.<sup>[5](https://pws.npru.ac.th/sartthong/data/files/Matrix%20Algebra%20theory%20computations%20and%20applications%20in%20statistics.pdf)</sup> In statistics, the least-squares hat matrix \( P = X(X^{\mathrm{T}}X)^{-1}X^{\mathrm{T}} \), which projects a response vector onto the column space of the design matrix \( X \), is idempotent, as is the residual-forming matrix \( R = I - P \); applying either twice has the same effect as applying it once.<sup>[12](https://statproofbook.github.io/P/mlr-idem)</sup><sup> • </sup><sup>[13](https://uchicago-pol-methods.github.io/EstimationI/ch01-review.html)</sup> The projection onto the orthogonal complement is obtained by subtracting from the identity, \( P_{\mathcal{S}^\perp} = I - P_{\mathcal{S}} \), mirroring the complement idempotent \( 1 - e \) in [ring theory](https://www.edgepedia.org/ring-theory).<sup>[14](https://stat151a.berkeley.edu/fall-2024/lectures/Lecture4.html)</sup>

## Related notions and current work

Idempotents contrast with nilpotent elements, whose some power is 0: an idempotent stabilizes after one multiplication, while a nilpotent element need not define any projection. Idempotents encode significant information about the algebraic structure of a ring, and they remain an active research object: recent work studies the partially ordered set of idempotents in matrix rings over division rings,<sup>[15](https://arxiv.org/abs/2510.09501)</sup> idempotent graphs of matrix rings,<sup>[16](https://arxiv.org/abs/2402.11256)</sup> and central idempotents in group-graded rings, where a nonzero central idempotent in a group ring \( A[G] \) has finite support group, a result due to Burns that generalizes to broad classes of graded rings.<sup>[17](https://arxiv.org/abs/2605.20008)</sup>

## References

1. [Algebra II Student Notes, RES.18-012 (Spring 2022), MIT OpenCourseWare](https://ocw.mit.edu/courses/res-18-012-algebra-ii-student-notes-spring-2022/mit18_702s22_full_lec.pdf)
2. [Romyar Sharifi, Abstract Algebra, Chapter 13: Representation theory, UCLA](https://www.math.ucla.edu/~sharifi/notes/algebra-ch13.html)
3. [Jason Starr, MAT 535 Algebra II, Stony Brook University](https://www.math.stonybrook.edu/~jstarr/M535s26/M535s26field.pdf)
4. [The Stacks Project, Section 10.21: Open and closed subsets of spectra](https://stacks.math.columbia.edu/tag/04PN)
5. [Matrix Algebra: Theory, Computations, and Applications in Statistics, Section 8.5](https://pws.npru.ac.th/sartthong/data/files/Matrix%20Algebra%20theory%20computations%20and%20applications%20in%20statistics.pdf)
6. [Chapter 1 Preliminaries, Methods of Ring Theory, ScienceDirect](https://www.sciencedirect.com/science/article/pii/S0304020808701981)
7. [J. Kettinger, Finite-Dimensional Algebras, University of Nebraska–Lincoln](https://www.math.unl.edu/~jkettinger2/Finite-Dimensional%20Algebras.pdf)
8. [Rings and algebras, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Rings_and_algebras)
9. [Ring Theory: Ideals of a ring, Noetherian rings and factorizations, Numericana](https://www.ww2000w.numericana.com/answer/rings.htm)
10. [Commutative Algebra lecture notes, Section 15: Open and closed subsets of spectra, CiteSeerX](https://citeseerx.ist.psu.edu/document?doi=fccb92ca6dd3edb2ad17f17ac92ff628fee42ce5&repid=rep1&type=pdf)
11. [Arvind Saibaba, Lectures on Algebra, Tata Institute of Fundamental Research](https://mathweb.tifr.res.in/~arvind/algebra/Lectures_algebra.pdf)
12. [Projection matrix and residual-forming matrix are idempotent, The Book of Statistical Proofs](https://statproofbook.github.io/P/mlr-idem)
13. [Probability and Linear Algebra, Estimation I: Computational Companion, University of Chicago](https://uchicago-pol-methods.github.io/EstimationI/ch01-review.html)
14. [Least squares as a projection, STAT 151A Lecture 4, UC Berkeley](https://stat151a.berkeley.edu/fall-2024/lectures/Lecture4.html)
15. [A Note on Idempotent Matrices: The Poset Structure and The Construction, arXiv](https://arxiv.org/abs/2510.09501)
16. [On the Idempotent Graph of Matrix Ring, arXiv](https://arxiv.org/abs/2402.11256)
17. [Central idempotents in group-graded rings, arXiv](https://arxiv.org/abs/2605.20008)

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