Pseudo-Inverses in Associative Rings and Semigroups

DOI10.2307/2308576zbMath0083.02901OpenAlexW4241143925WikidataQ29040072 ScholiaQ29040072MaRDI QIDQ3255427

Michael P. Drazin

Publication date: 1958

Published in: The American Mathematical Monthly (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.2307/2308576

zbMATH Keywords

rings, modules, fields

Related Items

The Drazin inverse in multibody system dynamics, Determinantal representations of the Drazin inverse over the quaternion skew field with applications to some matrix equations, The representation of the Drazin inverse of anti-triangular operator matrices based on resolvent expansions, On a result of J.J. Sylvester, The weighted \(*\)-core-nilpotent decomposition, Some results on the Drazin invertibility and idempotents, Characterizations and representations of the Drazin inverse involving idempotents, Pseudo-consimilarity and semi-consimilarity of complex matrices, Gröbner basis computation of Drazin inverses with multivariate rational function entries, The Drazin inverse of a semilinear transformation and its matrix representation, Generalized inverses of morphisms with kernels, Weighted generalized Drazin inverse in rings, Fredholm spectra and Weyl type theorems for Drazin invertible operators, Generalized B-Fredholm Banach algebra elements, Partial orders based on inverses along elements, Spectral properties of degenerate systems of ordinary differential equations on a finite interval, Weighted partial isometries and weighted-EP elements in \(C^{*}\)-algebras, A generalized inverse method for asymptotic linear programming, Representations for the Drazin inverses of \(2 \times 2\) block matrices, Multiscale analysis in many-body systems, The Drazin inverse of the sum of two matrices and its applications, The one-sided inverse along an element in semigroups and rings, Additive preservers of Drazin invertible operators with bounded index, Some results on the \((2, 2, 0)\) Drazin inverse problem, On Drazin inverse of operator matrices, A Drazin inverse for rectangular matrices, Local spectral theory for Drazin invertible operators, Additive results of the Drazin inverse for bounded linear operators, Inner inverses and inner annihilators in rings., On the Drazin inverse of the sum of two matrices, New characterizations for core inverses in rings with involution, On additive properties of the Drazin inverse of block matrices and representations, The representation and approximation of the Drazin inverse of a linear operator in Hilbert space., On generalized inverses and Green's relations, Drazin inverses and canonical forms in \(M_ 0(\)Z/h), On the Drazin inverses involving power commutativity, New additive results on Drazin inverse and its applications, Formulas for the Drazin inverse of matrices with new conditions and its applications, The representation and approximation of the \(W\)-weighted Drazin inverse of linear operators in Hilbert space., New characterizations of EP, generalized normal and generalized Hermitian elements in rings, Group inverses of matrices over right Ore domains, Periodic properties of groupbound semigroups, Nonnegative integral generalized inverses, Semisimilarity for matrices over a division ring, Note on an extension of the Moore-Penrose inverse, Perturbation bounds for Markov chains with general state space, Pseudo Drazin inverses in associative rings and Banach algebras, Strongly preserver problems in Banach algebras and \(C^{\ast}\)-algebras, Representations for the Drazin inverses of the sum of two matrices and some block matrices, Characterizations and representations of the inverse along an element, Jacobson's lemma for the generalized Drazin inverse, Positivity of discrete singular systems and their stability: an LP-based approach, A note on the representation for the Drazin inverse of \(2 \times 2\) block matrices, Extremal definitions of generalized inverses, Left and right generalized inverses, Drazin spectrum of operator matrices on the Banach space, Additive and product properties of Drazin inverses of elements in a ring, Stability analysis of positive descriptor systems, Representations for the pseudo Drazin inverse of elements in a Banach algebra, The reverse order law of the \((b,c)\)-inverse in semigroups, Reverse order laws for the Drazin inverses, Additive results for the Drazin inverse of block matrices and applications, Nonlinear preserver problems on \(\mathcal{B}(H)\), Continuity properties of the Drazin pseudoinverse, On certain finite-dimensional algebras generated by two idempotents, Spectral sets and the Drazin inverse with applications to second order differential equations., Representation for the \(W\)-weighted Drazin inverse of linear operators, Block generalized inverses, New additive results for the generalized Drazin inverse, Generalized Browder's and Weyl's theorems for left and right multiplication operators, Random walks and flights over connected graphs and complex networks, The representations of the Drazin inverse of differences of two matrices, Weyl type theorems for left and right polaroid operators, A generalization of strongly preserver problems of Drazin invertibility, Reverse order law for the core inverse in rings, Additive Drazin inverse preservers, \(B\)-Fredholm and spectral properties for multipliers in Banach algebras, Weak Drazin inverses, The Drazin inverses of sum and difference of idempotents, Some additive results on Drazin inverses, Spectral inverses and the row-space equations, A note on the perturbation of an outer inverse, Nonnegative mth roots of nonnegative 0-symmetric idempotent matrices, An index classification of M-matrices, On the \((p,q)\) outer generalized inverse in Banach algebra, Generalized inverse-positivity and splittings of M-matrices, Expression of the Drazin and MP-inverse of partitioned matrix and quotient identity of generalized Schur complement, EP elements in rings., The generalized Drazin inverse with commutativity up to a factor in a Banach algebra, Semistable operators and singularly perturbed differential equations, On the Drazin index of regular elements, On the Drazin inverse of block matrices and generalized Schur complement, Solvability, controllability, and observability of singular systems, On the computation of the transfer function matrix of singular systems, EP morphisms, A partial order in completely regular semigroups, Variation of the parameters formula and the problem of BIBO for singular measure differential systems with impulse effect, Concise row-pruning algorithm to invert a matrix, An alternative limit expression of Drazin inverse and its application, A weighted Drazin inverse and applications, Some properties of (b, c)-inverses in rings, A new class of generalized inverses in semigroups and rings with involution, On the coincidence of the Drazin inverse and the Drazin-Moore-Penrose inverses, Weak group inverses and partial isometries in proper *-rings, Weak core inverses and pseudo core inverses in a ring with involution, Unnamed Item, Unnamed Item, The W -weighted BT inverse, Pseudo-generalized inverse and Drazin invertibility, Jacobson's lemma for generalized Drazin-Riesz inverses, EP-like properties of \((b, c)\)-inverses, On perturbation of Drazin invertible linear relations, A class of outer generalized inverses and a note on Drazin’s finiteness conditions in rings and semigroups, Drazin inverse and generalization of core-nilpotent decomposition, Fixed-time convergence integral-enhanced ZNN for calculating complex-valued flow matrix Drazin inverse, Weighted generalized Moore-Penrose inverse, Unnamed Item, Further characterizations of the CMP inverse of matrices, Common properties among various generalized inverses and constrained binary relations, Intermixing pairs of generalized inverses, Reverse order law and forward order law for the (b, c)-inverse, Common properties of \(a\) and \(b\) satisfying \(ab^n = b^{n+1}\) and \(ba^n = a^{n+1}\) in Banach algebras, Drazin and group invertibility in algebras spanned by two idempotents, The polarity along an element in rings, On the explicit computation of the set of 1-inverses of a square matrix, Further results on weighted core inverse in a ring, Left and right-Drazin inverses in rings and operator algebras, Drazin invertibility of linear operators on quaternionic Banach spaces, One-sided w-core inverses in rings with an involution, New additive results for the Drazin inverse of multivalued operators, Characterizations and representations of weak core inverses and $m$-weak group inverses, Left \(w\)-core inverses in rings with involution, Generalized core-EP inverse for square matrices, The \((b, c)\)-core inverse and its dual in semigroups with involution, On the structure of semigroups whose regular elements are completely regular, Matrix derivatives and Kronecker products for the core and generalized core inverses, Generalized inverse equations and SEP elements in a ring with involution, \(B\)-Weyl and Drazin invertible operators linked by weak pseudo \(S_0\)-demicompactness, Unnamed Item, Unnamed Item, Remarks on the kernel of a matrix semigroup, Central Drazin inverses, Further results on the reverse order law for the group inverse in rings, Some additive results on Drazin inverse, Outer Generalized Inverses in Rings, Additive results for the generalized Drazin inverse in a Banach algebra, Characterizations for the n-strong Drazin invertibility in a ring, Some additive results on Drazin inverse, Generalized WG inverse, The generalized Drazin inverse of operator matrices, The $(j,m)$-core inverse in rings with involution, The representation and approximation for Drazin inverse, Inner actions of weak Hopf algebras, A note on Cline’s formula for the generalized Drazin inverse, Common spectral properties of linear operators A and B satisfying AkBkAk = Ak+1 and BkAkBk = Bk+1, Additive property of Drazin invertibility of elements in a ring, Stable perturbation in Banach algebras, Generalized Jacobson's lemma for Drazin inverses and its applications, The Drazin inverse of the sum of four matrices and its applications, Symbolic computation of Drazin inverses by specializations, Unnamed Item, New characterizations of the CMP inverse of matrices, Weighted (b,c)-inverses in categories and semigroups, A note on the formulas for the Drazin inverse of the sum of two matrices, Nonlinear maps preserving Drazin invertible operators of bounded index, The characterizations of one-sided generalized inverses, Elements of rings and Banach algebras with related spectral idempotents, Generalized inverses and their relations with clean decompositions, Right core inverse and the related generalized inverses, A completion theorem for finitely generated commutative semigroups, The existence and representation of the Drazin inverse of a 2 × 2 block matrix over a ring, Jacobson pairs and Bott-Duffin decompositions in rings, On the Drazin inverse of finite potent endomorphisms, Compact perturbations of Drazin invertible operators, On a question of Drazin, On Drazin-Moore-Penrose inverses of finite potent endomorphisms, Reverse Order Law for the Group Inverse in Semigroups and Rings, On (b,c)-inverses and (c,b)-inverses, EP properties of (b, c)-invertible matrices, Reverse order laws for Hirano inverses in rings, Rank equalities related to a class of outer generalized inverse, A new characterization of generalized Browder’s theorem and a cline’s formula for generalized Drazin-eromorphic inverses, On the weighted pseudo Drazin invertible elements in associative rings and Banach algebras, The outer inverse f(2)T,S of a homomorphism of right R-modules, One-sided (b,c)-inverses in rings, Idempotents generated by weighted generalized inverses in rings with involution, Explicit solutions of infinite linear systems associated with group inverse endomorphisms, ADDITIVE PROPERTIES OF THE DRAZIN INVERSE FOR MATRICES AND BLOCK REPRESENTATIONS: A SURVEY, Weighted weak core inverse of operators, Explicit solutions of non-homogeneous difference equations from finite potent endomorphisms, Weighted MPCEP inverse of an operator between Hilbert spaces, Some formulas on the Drazin inverse for the sum of two matrices and block matrices, Extensions of Jacobson's lemma for Drazin inverses, Displacement rank of the Drazin inverse, Exterior Dirichlet and Neumann problems and the linked ergodic inverse problems in the entire space, Drazin-Moore-Penrose invertibility in rings, The Drazin inverse of bounded operators with commutativity up to a factor, Revisiting of the BT-inverse of matrices, The Drazin inverses of products and differences of orthogonal projections, Linear boundary-value problems described by Drazin invertible operators, Weighted pre-orders in a Banach algebra, On the p-invertibility in rings and Banach algebras, Characterizations and maximal classes of elements related to pseudo core inverses, The weak core inverse, Moore-Penrose inverses in rings and weighted partial isometries in \(C^\ast \)-algebras, T-Jordan canonical form and T-Drazin inverse based on the T-product, Outer inverses and Jacobi type identities, Reverse order laws on the conditions of the commutativity up to a factor, Further characterizations of the weak group inverse of matrices and the weak group matrix, Representations for the Drazin inverse of the sum of two matrices and its applications, Reprint of: On generalized inverses and Green's relations, Wg-Drazin-star operator and its dual, A class of outer generalized inverses, Generalized pseudoskeleton decompositions, Elementary, finite and linear vN-regular cellular automata, On generalized Drazin invertible linear relations, Perturbation theory for the Fokker-Planck operator in chaos, On matrices over an arbitrary semiring and their generalized inverses, The Drazin inverse of a singular, unreduced tridiagonal matrix, On the Drazin inverse of the linear combinations of two idempotents in a complex Banach algebra, Moore-Penrose invertibility of differences and products of projections in rings with involution, Semistability theory for spatially distributed systems, An analog of the adjugate matrix for the outer inverse \(A^{(2)}_{T, S}\), Further results on weighted core-EP inverse of matrices, \(m\)-weak group inverses in a ring with involution, Stability of essential B-spectra of unbounded linear operators and applications, On the continuity of the group inverse in \(C^*\)-algebras, Generalized inverses: uniqueness proofs and three new classes, A matrix iteration for finding Drazin inverse with ninth-order convergence, An improved computationally efficient method for finding the Drazin inverse, Perturbation of the Drazin inverse for closed linear operators, On Hartwig-Nambooripad orders, Reverse order laws for the generalized strong Drazin inverses, The perturbation theory for the Drazin inverse and its applications II, An algebraic interpretation of cointegration, Generalized inverses, Centralizer's applications to the inverse along an element, A note on the generalized Drazin-Riesz invertible operators, An approach to quasipolarity for rings along nilpotent elements, New extensions of Jacobson's lemma and Cline's formula, EP-quasinilpotent decomposition and its generalizations, Drazin-star and star-Drazin inverses of bounded finite potent operators on Hilbert spaces, GENERALIZATIONS OF FITTING'S LEMMA IN ARBITRARY ASSOCIATIVE RINGS, Centralizers and their applications to generalized inverses, On finite element method-flux corrected transport stabilization for advection-diffusion problems in a partial differential-algebraic framework, Generalized inverses and the total stopping times of collatz sequences, Strong stability and perturbation bounds for discrete Markov chains, The \((2,2,0)\) Drazin inverse problem, Drazin inverse conditions for stability of positive singular systems, Partial orders in rings based on generalized inverses -- unified theory., Additive perturbation results for the Drazin inverse, Generalized Browder's and Weyl's theorems for generalized derivations, Index of B-Fredholm operators and generalization of a Weyl theorem, Drazin spectra of Banach space operators and Banach algebra elements, Characterizations and perturbation analysis of a class of matrices related to core-EP inverses, The Drazin inverses of infinite triangular matrices and their linear preservers, Condition numbers and structured perturbation of the \(W\)-weighted Drazin inverse, Further results on the inverse along an element in semigroups and rings, Core-nilpotent endomorphisms of infinite-dimensional vector spaces, Generalized inverses of elements and their polarities in rings, Outer inverses: characterization and applications, Projections and isolated points of parts of the spectrum, A note on computational formulas for the Drazin inverse of certain block matrices, Group inverse for two classes of \(2\times 2\) block matrices over rings, Rakočević's property for generalized derivations, On the Drazin inverse of the sum of two operators and its application to operator matrices, A note on Koliha-Drazin invertible operators and a-Browder's theorem, Weighted extended \(g\)-Drazin inverse, On the matrix equation \(Ax =\lambda Bx\), Core and core-EP inverses of tensors, Extended \(g\)-Drazin inverse in a Banach algebra, On Drazin spectral equation for the operator products, On identity bases of epigroup varieties, The inverse along an element in rings, Centralizer's applications to the \((b, c)\)-inverses in rings, Weighted CMP inverse of an operator between Hilbert spaces, Left core inverses in rings with involution, On the generalized Drazin-Riesz inverse for closed linear operators, Approximate preservers on Banach algebras and \(C^*\)-algebras, Drazin invertibility for matrices over an arbitrary ring, The Drazin and generalized Drazin invertibility of linear combinations of idempotents, The Souriau-Frame algorithm and the Drazin pseudoinverse, A weak group inverse for rectangular matrices, The Drazin inverse and a spectral inequality of Marcus, Minc, and Moyls, Properties of core-EP order in rings with involution, B-Weyl spectrum and poles of the resolvent, A characterization for the \(W\)-weighted Drazin inverse and a Cramer rule for the \(W\)-weighted Drazin inverse solution, New results on common properties of the products \(AC\) and \(BA\), A finite algorithm for the Drazin inverse of a polynomial matrix, Reverse order laws for the generalized Drazin inverse in Banach algebras, *-DMP elements in *-semigroups and *-rings, MORE ON THE NON-COMMUTATIVE EPIGROUPS WHICH ARE NOT QUASI-COMMUTATIVE, Unnamed Item, THE LINK BETWEEN REGULARITY AND STRONG-PI-REGULARITY, Elements of rings with equal spectral idempotents, Generalized Drazin Invertibility of Operator Matrices, Some new results on the core partial order, Further results on partial ordering and the generalized inverses, Unnamed Item, Strong regularity and generalized inverses in jordan systems, On the History of Differential-Algebraic Equations, Unnamed Item, An Application of the Moore-Penrose Inverse to Antisymmetric Relations, One-sided central Drazin inverses, Generalized Weyl's theorem and hyponormal operators, The pseudo core inverses of differences and products of projections in rings with involution, On -based extensions of the sharp order, Generalizations of some conditions for Drazin inverses of the sum of two matrices, On a Theorem of Flanders, A note on clean elements and inverses along an element, Jacobson's lemma for the generalized \(n\)-strong Drazin inverses in rings and in operator algebras, On G-Drazin inverses of finite potent endomorphisms and arbitrary square matrices, Unnamed Item, The generalized inverses of the products of two elements in a ring, On elements whose Moore–Penrose inverse is idempotent in a*-ring, Properties of generalized strongly Drazin invertible elements in general rings, The (b,c)-inverse for products and lower triangular matrices, Inner, outer, and generalized inverses in banach and hilbert spaces, The Generalized Inverse A⁽²⁾_{T, S}of a Matrix Over an Associative Ring, Reverse order law for the inverse along an element, Matrix Semigroups, On Drazin invertibility, Analogs of the adjoint matrix for generalized inverses and corresponding Cramer rules, Unnamed Item, Jacobson’s lemma and Cline’s formula for generalized inverses in a ring with involution, Maximal classes of operators determining some weighted generalized inverses, Core-Nilpotent decomposition and new generalized inverses of finite potent endomorphisms, Special clean elements in rings, ∗-Strongly regular rings, Weak group inverses in proper ∗-rings, An Efficient Factorization for the Group Inverse, Generalized Drazin Inverse and Commuting Riesz Perturbations, Left and right generalized Drazin invertible operators and local spectral theory, Characterizations for pseudo core inverses in a ring with involution, On DMP inverses and m-EP elements in rings, An extension of Clines formula for a generalized Drazin inverse, Extensions of quasipolar rings, Some characterizations of rightc-regularity and(b;c)-inverse, Weighted pseudo core inverses in rings, A generalized Drazin inverse, Characterizations of clean elements by means of outer inverses in rings and applications, Weak inverses of products — Cline’s formula meets Jacobson lemma, Further Results on the Sharp Ordering in Rings, Drazin Invertibility in a Certain Finite-Dimensional Algebra Generated by Two Idempotents, Subclasses of (b, c)-inverses, Orders in rings based on the core-nilpotent decomposition, Pseudo core inverses in rings with involution, Further results on several types of generalized inverses, Absorption laws and reverse order laws for generalized core inverses, Unnamed Item, A Generalized Matrix Inverse That Is Consistent with Respect to Diagonal Transformations, Characterizations of m-EP elements in rings, Generalized invertibility in a corner ring, The Drazin inverse of the sum of two bounded linear operators and it’s applications, The Drazin spectrum of tensor product of Banach algebra elements and elementary operators, New extensions of Cline’s formula for generalized inverses, On the pseudo drazin inverse of the sum of two elements in a Banach algebra, Representations of the (b,c)-inverses in rings with involution, Weighted core–EP inverse of an operator between Hilbert spaces, Normal form and representation theory, Solution Concepts for Linear DAEs: A Survey, Unnamed Item, Unnamed Item, A class of Drazin inverses in rings, The one-sided inverse along two elements in rings, On Linear Algebraic Semigroups, Unnamed Item, The generalized and pseudo n-strong Drazin inverses in rings, Eventually regular semigroups, On the Drazin inverse and M-P inverse for sum of matrices, Theory and applications of matrix perturbations, with respect to Hankel matrices and power formulae, THE DRAZIN INVERSES OF PRODUCTS AND DIFFERENCES OF PROJECTIONS IN A C^*-ALGEBRA, The Drazin inverse in an arbitrary semiring, Exchange rings, exchange equations, and lifting properties, Determining Jordan (triple) homomorphisms by invertibility preserving conditions, Left and right generalized Drazin invertible operators, Generalized inverses of matrices: a perspective of the work of Penrose, Semigroups satisfying minimal conditions II, A note on additive results for the Drazin inverse, Further results on common properties of the products ac and bd, On Jacobson's lemma and Cline's formula for Drazin inverses, Generalized inverses modulo ℋ in semigroups and rings, EP elements and *-strongly regular rings, Unnamed Item, Ordering Orders and Quotient Rings, Commuting properties of generalized inverses, Counting Matrices by Drazin Index, The commuting inverse along an element in semigroups, Additive results for the generalized Drazin inverse, On some orders in ∗-rings based on the core-EP decomposition