Definition:Subring
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Definition
Let $\left({R, +, \circ}\right)$ be an algebraic structure with two operations.
A subring of $\left({R, +, \circ}\right)$ is a subset $S$ of $R$ such that $\left({S, +_S, \circ_S}\right)$ is a ring.
Proper Subring
A subring $S$ of $R$ is said to be a proper subring iff $S$ is not the null ring nor $R$ itself.
Also see
Notes
Some sources insist that $R$ must be a ring for $S$ to be definable as a subring, but this limitation is unnecessarily restricting.
Sources
- Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.2$
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 21$
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 5.19$
- B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 2.1$: Definition $2.1$
- Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 56$: Definition $1$
