%path = "maths/stuctures/ring" %kind = kinda["texts"] %level = 10
Ring-like algebraic structures build on top of group-like structures ({{!util.a("r.cl")}}) and consist of a set \(M\) with two binary operations \(+\) and \(\cdot\) , in short \((M,+,\cdot)\).
\((M,+)\) and \((M,\cdot)\) are monoids and \(0\cdot a = 0\) holds \(\rightarrow\) Semiring.
\((M,+)\) is commutative group \(\rightarrow\) Ring.
In \((M,\cdot)\) there are no two numbers, whose product is 0. Free of zero divisor \(\rightarrow\) Integral domain.
\((M\setminus\{0\},\cdot)\) is a commutative Group \(\rightarrow\) Field.
\((M,\cdot)\) satisfies the Jakobi Identity \(a\cdot (b \cdot c) + c\cdot (a \cdot b) + b\cdot (c \cdot a) = 0\) \(\rightarrow\) Lie Ring.
\((M,\cdot)\) is idempotent \(\rightarrow\) Boolean Algebra.