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Let $A= \bigoplus_{i=0}^{N}A_i$ be a finitely graded ring with the property that an element $(a,b)$ is zero under the natural map $$A_i \times A_j \to A_{i+j}$$ if and only if $i+j > N$ or $a=b=0$. Hence this ring is "almost" an integral domain until degree $N+1$.

Do these types of rings have a name?

If we assume $A$ is noetherian then it must be a finitely generated algebra over the noetherian ring $A_0$ and thus the quotient $R/I$ of a polynomial ring $R$ over $A_0$.

Are there any necessary and sufficient conditions we may impose on $I$ so that the property above holds?

Let $A= \bigoplus_{i=0}^{N}A_i$ be a finitely graded ring with the following property: if $x \in A_i$ and $y \in A_j$ and $i+j \leq N$, then $$xy = 0 \text{ implies } x = 0 \text{ or } y = 0.$$ Hence this ring is "almost" an integral domain until degree $N+1$.

Do these types of rings have a name?

If we assume $A$ is noetherian then it must be a finitely generated algebra over the noetherian ring $A_0$ and thus the quotient $R/I$ of a polynomial ring $R$ over $A_0$.

Are there any necessary and sufficient conditions we may impose on $I$ so that the property above holds?

Let $A= \bigoplus_{i=0}^{N}A_i$ be a finitely graded ring with the property that whenever an element $(a,b)$ is zero under the natural map $$A_i \times A_j \to A_{i+j}$$ if and only if $i+j > N$ or $a=b=0$. Hence this ring is "almost" an integral domain until degree $N+1$.

Do these types of rings have a name?

If we assume $A$ is noetherian then it must be a finitely generated algebra over the noetherian ring $A_0$ and thus the quotient $R/I$ of a polynomial ring $R$ over $A_0$.

Are there any necessary and sufficient conditions we may impose on $I$ so that the property above holds?

Let $A= \bigoplus_{i=0}^{N}A_i$ be a finitely graded ring with the property that an element $(a,b)$ is zero under the natural map $$A_i \times A_j \to A_{i+j}$$ if and only if $i+j > N$ or $a=b=0$. Hence this ring is "almost" an integral domain until degree $N+1$.

Do these types of rings have a name?

If we assume $A$ is noetherian then it must be a finitely generated algebra over the noetherian ring $A_0$ and thus the quotient $R/I$ of a polynomial ring $R$ over $A_0$.

Are there any necessary and sufficient conditions we may impose on $I$ so that the property above holds?

Let $A= \bigoplus_{i=0}^{N}A_i$ be a finitely graded ring with the property that whenever an element $a \otimes b$ is zero under the natural map $$A_i \otimes_{A_0} A_j \to A_{i+j}$$ if and only if $i+j > N$. Hence this ring is "almost" an integral domain until degree $N+1$.

Do these types of rings have a name?

If we assume $A$ is noetherian then it must be a finitely generated algebra over the noetherian ring $A_0$ and thus the quotient $R/I$ of a polynomial ring $R$ over $A_0$.

Are there any necessary and sufficient conditions we may impose on $I$ so that the property above holds?

Let $A= \bigoplus_{i=0}^{N}A_i$ be a finitely graded ring with the property that whenever an element $(a,b)$ is zero under the natural map $$A_i \times A_j \to A_{i+j}$$ if and only if $i+j > N$ or $a=b=0$. Hence this ring is "almost" an integral domain until degree $N+1$.

Do these types of rings have a name?

If we assume $A$ is noetherian then it must be a finitely generated algebra over the noetherian ring $A_0$ and thus the quotient $R/I$ of a polynomial ring $R$ over $A_0$.

Are there any necessary and sufficient conditions we may impose on $I$ so that the property above holds?