MHB What distinguishes a Field from a Ring?

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A field is a set equipped with two operations, addition and multiplication, where both operations are commutative, and every non-zero element has a multiplicative inverse. In contrast, a ring may not have commutative multiplication and does not require every element to have a multiplicative inverse. Some definitions of rings include a multiplicative identity, while others do not. Both structures share the same rules for addition. Understanding these distinctions is crucial for studying abstract algebra.
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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 'How to define a vector field?'

Thread 'How to define a vector field?'

Hello! In one book I saw that function ##V## of 3 variables ##V_x, V_y, V_z## (vector field in 3D) can be decomposed in a Taylor series without higher-order terms (partial derivative of second power and higher) at point ##(0,0,0)## such way: I think so: higher-order terms can be neglected because partial derivative of second power and higher are equal to 0. Is this true? And how to define vector field correctly for this case? (In the book I found nothing and my attempt was wrong...
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