MHB What distinguishes a Field from a Ring?

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What the differences between Field and Ring?
 
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A ring isn't necessarily commutative in its multiplication, and doesn't necessarily have multiplicative inverses. Some authors assume rings have a multiplicative identity $1$, other authors don't. Addition is the same.
 
Thread 'Derivation of equations of stress tensor transformation'

Thread 'Derivation of equations of stress tensor transformation'

Hello ! I derived equations of stress tensor 2D transformation. Some details: I have plane ABCD in two cases (see top on the pic) and I know tensor components for case 1 only. Only plane ABCD rotate in two cases (top of the picture) but not coordinate system. Coordinate system rotates only on the bottom of picture. I want to obtain expression that connects tensor for case 1 and tensor for case 2. My attempt: Are these equations correct? Is there more easier expression for stress tensor...
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