Definition:Generated Subring
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Definition
Let $\struct {R, +, \circ}$ be a ring.
Let $S \subseteq R$ be a subset.
The subring generated by $S$ is the smallest subring of $R$ containing $S$; that is, it is the intersection of all subrings of $R$ containing $S$.
Also see
- Intersection of Subrings is Subring
- Definition:Generator of Ring
- Definition:Generator of Algebra over Ring
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.3$: Some properties of subrings and ideals: Definition $2.14$