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1 Definitions and Examples of Morita Context Rings 1

1.1 Definitions of Morita context rings 1

1.2 Classical matrix algebras 7

1.2.1 Full matrix algebras 7

1.2.2 Triangular matrix algebras 7

1.2.3 Block upper triangular matrix algebras 8

1.2.4 Inflated algebras 9

1.3 Quasi-hereditary algebras 10

1.3.1 Basic construction 10

1.3.2 Dual extension algebras 11

1.4 Two non-degenerate examples 14

1.4.1 Morita context rings from smash product 14

1.4.2 Morita context rings from group algebras 15

1.5 Examples of operator algebras 17

1.5.1 Triangular Banach algebras 17

1.5.2 Nest algebras 17

1.5.3 von Neumann algebras 19

1.5.4 Incidence algebras 20

2 Linear Mappings on Morita Context Rings 23

2.1 Commuting mappings on Morita context rings 23

2.1.1 Posner Theorem 25

2.1.2 Commuting mappings and centralizing mappings 28

2.1.3 Skew commuting and skew centralizing mappings 49

2.2 Lie derivations on Morita context rings 62

2.3 Jordan derivations on Morita context rings 71

2.4 Jordan generalized derivations on triangular algebras 85

2.5 Lie triple derivations on triangular algebras 94

2.5.1 Proof of the main Theorem 95

2.5.2 Another look to Theorem 2.5.1 99

2.6 Local actions of linear mappings on Morita context rings 104

3 Non-linear Mappings and Higher Mappings 111

3.1 Characterization of Jordan higher derivations 111

3.2 Jordan higher derivations on some operator algebras 118

3.3 Jordan higher derivations on triangular algebras 123

3.4 When a higher derivation is inner 136

3.5 Non-linear Lie higher derivations 147

3.6 Non linear Jordan bijective mappings 163

3.7 Jordan higher derivable points 178

Bibliography 181