Generally, in any human field, a Smarandache Structure on a set A means a weak structure W on A such that there exists a proper subset B which is embedded with a stronger structure S.

By proper subset one understands a set included in A, different from the empty set, from the unit element if any, and from A.

These types of structures occur in our every day’s life, that’s why we study them in this book.

Thus, as two particular cases:

A Smarandache ring of level I (S-ring I) is a ring R that contains a proper subset that is a field with respect to the operations induced.

A Smarandache ring of level II (S-ring II) is a ring R that contains a proper subset A that verifies:

· A is an additive abelian group;

· A is a semigroup under multiplication;

· For a, b in A, a·b = 0 if and only if a = 0 or b = 0.

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