Algebraic properties describe the rules that apply to a set of numbers (such as those described in the real number system) and an operation, such as addition, subtraction, multiplication, or division.  For example, the commutative property applies to integers for the operation of addition. 

Here are some algebraic properties:

• Commutative property:  if the commutative property applies to a set, then the order in which we apply the operation does not matter.  For example, the commutative property applies to integers for the operation of addition:  a + b = b + a
• Associative property:  if the associative property applies to a set, then the grouping of elements does not affect the outcome of the operation.  For example, the associative property applies to integers for the operation of addition:  a + (b + c) = (a + b) + c
• Identity property:  if the identity property applies to a set, then there exists an element that we can apply the operation to and get the element we started with (the identical value).  For example, the identity element for integers under the operation of addition is 0:  a + 0 = a
• Inverse property:   if the inverse property applies to a set, then there exists an element in the set which when the operation is applied gives us the identity element.  For example, if we are looking at the set of integers under the operation of addition, then the opposite of the element, when added, gives us the identity element of 0:  a + (-a) = 0

There are many other properties:  the distributive property, properties of equality, and properties of inequality name just a few.