Contents1 Algebraic Basic Knowledge 11.1 Complex Matrices 11.1.1 Some Decompositions of Complex Matrices 11.1.2 The Index of a Square Complex Matrix 51.1.3 Idempotents�� Projections and EP Matrices 61.2 Definitions and Examples of Rings 71.2.1 Basic Concepts and Examples 71.2.2 Some Extensions of Rings 91.2.3 Idempotents�� Units and Regular Elements 121.2.4 One-Sided Invertibility and Invertibility 141.3 Semigroups�� Rings and Categories with Involution 161.3.1 Definitions and Examples 161.3.2 Proper Involutions 201.3.3 The Gelfand-Naimark Property 211.4 Regularity and -Regularity of Rings 211.4.1 Regularity and FP-Injectivity 211.4.2 Regularity 241.5 Invertibility of the Difference and the Sum of Idempotents 281.5.1 Invertibility of the Difference of Idempotents 281.5.2 Invertibility of the Sum of Idempotents 312 Moore-Penrose Inverses 352.1 Moore-Penrose Inverses of Complex Matrices 362.2 Characterizations of Moore-Penrose Inverses of Elements in Semigroups or Rings 382.2.1 Moore-Penrose Inverses of Elements in a Semigroup 392.2.2 Moore-Penrose Inverses of Elements in a Ring 422.2.3 Moore-Penrose Inverses of Matrices over a Ring 492.3 The Moore-Penrose Inverse of a Product 552.3.1 The Moore-Penrose Inverse of a Product paq 552.3.2 The Moore-Penrose Inverse of a Matrix Product 612.4 Moore-Penrose Inverses of Differences and Products of Projections 642.5 Jacobson��s Lemma for Moore-Penrose Inverses 722.5.1 Jacobson��s Lemma for Moore-Penrose Inverses in a (Generalized) GN Ring 722.5.2 Jacobson��s Lemma for Moore-Penrose Inverses in a Ring 742.6 The Moore-Penrose Inverse of a 2��2 Block Matrix 782.7 The Moore-Penrose Inverse of a Companion Matrix 852.8 The Moore-Penrose Inverse of a Sum of Morphisms 893 Group Inverses 973.1 Group Inverses of Complex Matrices 973.2 Characterizations of Group Inverses of Elements in Semigroups and Rings 983.3 The Group Inverse of a Product paq 1023.4 The Group Inverse of a Sum of Morphisms 1053.5 The Group Inverse of the Sum of Two Group Invertible Elements 1093.6 The Group Inverse of the Product of Two Regular Elements 1173.7 Group Inverses of Block Matrices 1223.8 Group Inverses of Companion Matrices Over a Ring 1283.9 EP Elements 1344 Drazin Inverses 1434.1 Drazin Inverses of Complex Matrices 1434.2 Drazin Inverses of Elements in Semigroups and Rings 1464.3 Drazin Invertibility in Two Semigroups of a Ring 1504.4 Jacobson��s Lemma and Cline��s Formula for Drazin Inverses 1564.5 Additive Properties of Drazin Inverses of Elements 1614.6 Drazin Inverses of Products and Differences of Idempotents 1654.7 Drazin Inverses of Matrices Over a Ring 1704.8 The Drazin Inverse of a Sum of Morphisms 1815 Core Inverses 2095.1 Core Inverses of Complex Matrices 2105.2 Core Inverses of Elements in Rings with Involution 2115.2.1 Equivalent Definitions and Characterizations of Core Inverses 2125.2.2 Relationship Between Group Invertibility and Core Invertibility 2145.2.3 Characterizations of Core Inverses by Algebraic Equations 2165.3 Characterizations of Core Invertibility by Special Elements 2225.3.1 Characterizations of Core Invertibility by Hermitian Elements or Projections in a Ring 2225.3.2 Characterizations of Core Invertibility for a Regular Element by Units in a Ring 2275.4 The Core Inverse of the Sum of Two Core Invertible Elements 2315.5 The Core Inverse of a Product paq 2425.6 Core Inverses of Companion Matrices 2475.7 The Core Inverse of a Sum of Morphisms 2536 Pseudo Core Inverses 2656.1 Core-EP Inverses of Complex Matrices 2656.2 Pseudo Core Inverses of Elements in Rings with Involution 2716.3 Additive and Multiplicative Properties 2826.4 Pseudo Core Inverses of Jacobson Pairs 2856.5 Pseudo Core Inverses of ab and ba 2906.6 The Pseudo Core Inverse of a Sum of Morphisms 2936.7 The Pseudo Core Inverse of a Product 3026.8 The Pseudo Core Inverse of a Low Triangular Matrix 307References 315Index 321